metric tensor:
$g_{ab}$

metric as background metric plus perturbation:
$g_{ab} = \hat{g}_{ab} + h_{ab}$

metric partial:
$g_{ab,c} = \hat{g}_{ab,c} + h_{ab,c}$

Levi-Civita connection of the background metric:
$\hat{\Gamma}_{abc} = \frac{1}{2} (\hat{g}_{ab,c} + \hat{g}_{ac,b} - \hat{g}_{bc,a})$
Levi-Civita connection of the metric:
$\Gamma_{abc} = \frac{1}{2} (\hat{g}_{ab,c} + \hat{g}_{ac,b} - \hat{g}_{bc,a} + h_{ab,c} + h_{ac,b} - h_{bc,a})$
$\Gamma_{abc} = \hat{\Gamma}_{abc} + \frac{1}{2} (h_{ab,c} + h_{ac,b} - h_{bc,a})$
Let $\Delta_{abc} = \frac{1}{2} (h_{ab,c} + h_{ac,b} - h_{bc,a})$
$\Gamma_{abc} = \hat{\Gamma}_{abc} + \Delta_{abc}$

metric inverse (using http://www.jstor.org/stable/2690437):
$g^{ab} = \hat{g}^{ab} - \frac{1}{g + 1} \hat{g}^{ac} h_{cd} \hat{g}^{db}$

weak field limit, assuming $\hat{g}_{ab} >> h_{ab}$, therefore $\hat{g}_{ab} - h_{ab} \approx \hat{g}_{ab}$:
$g^{ab} \approx \hat{g}^{ab}$

${\Gamma^a}_{bc} = \hat{g}^{ad} \hat{\Gamma}_{dbc} + \hat{g}^{ad} \Delta_{dbc} = {\hat{\Gamma}^a}_{bc} + {\Delta^a}_{bc}$

${\Gamma^a}_{bc,d} = {\hat{\Gamma}^a}_{bc,d} + {\Delta^a}_{bc,d}$
$= {\hat{\Gamma}^a}_{bc,d} + (\hat{g}^{ae} \Delta_{ebc})_{,d}$
$= {\hat{\Gamma}^a}_{bc,d} + {\hat{g}^{ae}}_{,d} \Delta_{ebc} + \hat{g}^{ae} \Delta_{ebc,d} $
$= {\hat{\Gamma}^a}_{bc,d} - \hat{g}^{af} \hat{g}_{fg,d} \hat{g}^{ge} \Delta_{ebc} + \frac{1}{2} \hat{g}^{ae} (h_{eb,c} + h_{ec,b} - h_{bc,e})_{,d} $
$= {\hat{\Gamma}^a}_{bc,d} - 2 \hat{g}^{af} \hat{\Gamma}_{(fg)d} \hat{g}^{ge} \Delta_{ebc} + \frac{1}{2} \hat{g}^{ae} (h_{eb,cd} + h_{ec,bd} - h_{bc,ed}) $
$= {\hat{\Gamma}^a}_{bc,d} - 2 {\hat{\Gamma}^{(ae)}}_d \Delta_{ebc} + \frac{1}{2} \hat{g}^{ae} (h_{eb,cd} + h_{ec,bd} - h_{bc,ed}) $

${\Gamma^a}_{bd,c} - {\Gamma^a}_{bc,d} = {\hat{\Gamma}^a}_{bd,c} - \hat{g}^{af} \hat{g}_{fg,c} \hat{g}^{ge} \Delta_{ebd} + \frac{1}{2} \hat{g}^{ae} (h_{eb,dc} + h_{ed,bc} - h_{bd,ec}) - {\hat{\Gamma}^a}_{bc,d} + \hat{g}^{af} \hat{g}_{fg,d} \hat{g}^{ge} \Delta_{ebc} - \frac{1}{2} \hat{g}^{ae} (h_{eb,cd} + h_{ec,bd} - h_{bc,ed}) $
$= {\hat{\Gamma}^a}_{bd,c} - {\hat{\Gamma}^a}_{bc,d} + \hat{g}^{af} \hat{g}^{ge} ( \hat{g}_{fg,d} \Delta_{ebc} - \hat{g}_{fg,c} \Delta_{ebd} ) + \frac{1}{2} \hat{g}^{ae} ( h_{ed,bc} - h_{bd,ec} - h_{ec,bd} + h_{bc,ed} ) $
Notice that, if you are using a cartesian background metric, then the first and second terms disappear.

${\Gamma^a}_{ec} {\Gamma^e}_{bd} = ({\hat{\Gamma}^a}_{ec} + {\Delta^a}_{ec}) ({\hat{\Gamma}^e}_{bd} + {\Delta^e}_{bd})$
$= {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\hat{\Gamma}^a}_{ec} {\Delta^e}_{bd} + {\Delta^a}_{ec} {\Delta^e}_{bd}$
Weak field limit, assume $(\partial h_{ab})^2 \approx 0$, therefore $(\Delta_{abc})^2 \approx 0$:
${\Gamma^a}_{ec} {\Gamma^e}_{bd} = {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\hat{\Gamma}^a}_{ec} {\Delta^e}_{bd}$
Notice that, if you are using a cartesian background metric, then the remaining terms disappear.

${R^a}_{bcd} = {\Gamma^a}_{bd,c} - {\Gamma^a}_{bc,d} + {\Gamma^a}_{ec} {\Gamma^e}_{bd} - {\Gamma^a}_{ed} {\Gamma^e}_{bc}$
$= {\hat{\Gamma}^a}_{bd,c} - {\hat{\Gamma}^a}_{bc,d} + \hat{g}^{af} \hat{g}^{ge} ( \hat{g}_{fg,d} \Delta_{ebc} - \hat{g}_{fg,c} \Delta_{ebd} ) + \frac{1}{2} \hat{g}^{ae} ( h_{ed,bc} - h_{bd,ec} - h_{ec,bd} + h_{bc,ed} ) + {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\hat{\Gamma}^a}_{ec} {\Delta^e}_{bd} - {\hat{\Gamma}^a}_{ed} {\hat{\Gamma}^e}_{bc} - {\Delta^a}_{ed} {\hat{\Gamma}^e}_{bc} - {\hat{\Gamma}^a}_{ed} {\Delta^e}_{bc} $
... write out $\hat{g}_{ab,c}$ in terms of $\hat{\Gamma}_{abc}$ ...
$= {\hat{\Gamma}^a}_{bd,c} - {\hat{\Gamma}^a}_{bc,d} + \hat{g}^{af} \hat{g}^{ge} ( (\hat{\Gamma}_{fgd} + \hat{\Gamma}_{gfd}) \Delta_{ebc} - (\hat{\Gamma}_{fgc} + \hat{\Gamma}_{gfc}) \Delta_{ebd} ) + \frac{1}{2} \hat{g}^{ae} ( h_{ed,bc} - h_{bd,ec} - h_{ec,bd} + h_{bc,ed} ) + {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\hat{\Gamma}^a}_{ec} {\Delta^e}_{bd} - {\hat{\Gamma}^a}_{ed} {\hat{\Gamma}^e}_{bc} - {\Delta^a}_{ed} {\hat{\Gamma}^e}_{bc} - {\hat{\Gamma}^a}_{ed} {\Delta^e}_{bc} $
...try to cancel some $\hat{\Gamma}_{abc} \Delta_{def}$ terms...
$= {\hat{\Gamma}^a}_{bd,c} - {\hat{\Gamma}^a}_{bc,d} + {\hat{\Gamma}^a}_{ed} {\Delta^e}_{bc} - {\hat{\Gamma}^a}_{ec} {\Delta^e}_{bd} + {\hat{\Gamma}^{ea}}_d \Delta_{ebc} - {\hat{\Gamma}^{ea}}_c \Delta_{ebd} + \frac{1}{2} \hat{g}^{ae} ( h_{ed,bc} - h_{bd,ec} - h_{ec,bd} + h_{bc,ed} ) + {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} + {\hat{\Gamma}^a}_{ec} {\Delta^e}_{bd} - {\hat{\Gamma}^a}_{ed} {\hat{\Gamma}^e}_{bc} - {\Delta^a}_{ed} {\hat{\Gamma}^e}_{bc} - {\hat{\Gamma}^a}_{ed} {\Delta^e}_{bc} $
...move those originally-$\partial \hat{g}_{ab}$ terms to the end, and cancel...
$= {\hat{\Gamma}^a}_{bd,c} - {\hat{\Gamma}^a}_{bc,d} + \frac{1}{2} \hat{g}^{ae} ( h_{ed,bc} - h_{bd,ec} - h_{ec,bd} + h_{bc,ed} ) + {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} - {\hat{\Gamma}^a}_{ed} {\hat{\Gamma}^e}_{bc} + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} - {\Delta^a}_{ed} {\hat{\Gamma}^e}_{bc} + {\hat{\Gamma}^{ea}}_d \Delta_{ebc} - {\hat{\Gamma}^{ea}}_c \Delta_{ebd} $

Let ${\hat{R}^a}_{bcd} = {\hat{\Gamma}^a}_{bd,c} - {\hat{\Gamma}^a}_{bc,d} + {\hat{\Gamma}^a}_{ec} {\hat{\Gamma}^e}_{bd} - {\hat{\Gamma}^a}_{ed} {\hat{\Gamma}^e}_{bc} $
Substitute to find:
${R^a}_{bcd} = {\hat{R}^a}_{bcd} + \frac{1}{2} \hat{g}^{ae} ( h_{ed,bc} - h_{bd,ec} - h_{ec,bd} + h_{bc,ed} ) + {\Delta^a}_{ec} {\hat{\Gamma}^e}_{bd} - {\Delta^a}_{ed} {\hat{\Gamma}^e}_{bc} + {\hat{\Gamma}^{ea}}_d \Delta_{ebc} - {\hat{\Gamma}^{ea}}_c \Delta_{ebd} $

Ricci curvature:
$R_{ab} = {R^c}_{acb}$
$= {\hat{R}^c}_{acb} + \frac{1}{2} \hat{g}^{ce} ( h_{eb,ac} - h_{ab,ec} - h_{ec,ab} + h_{ac,eb} ) + {\Delta^c}_{ec} {\hat{\Gamma}^e}_{ab} - {\Delta^c}_{eb} {\hat{\Gamma}^e}_{ac} + {\hat{\Gamma}^{ec}}_b \Delta_{eac} - {\hat{\Gamma}^{ec}}_c \Delta_{eab} $
Let ${\hat{R}^c}_{acb} = \hat{R}_{ab}$
$R_{ab} = \hat{R}_{ab} + \frac{1}{2} \hat{g}^{ce} ( h_{ac,eb} - h_{ab,ec} - h_{ec,ab} + h_{eb,ac} ) + {\Delta^c}_{ec} {\hat{\Gamma}^e}_{ab} - {\Delta^c}_{eb} {\hat{\Gamma}^e}_{ac} + {\hat{\Gamma}^{ec}}_b \Delta_{eac} - {\hat{\Gamma}^{ec}}_c \Delta_{eab} $

$R = \hat{g}^{ab} R_{ab}$
$= \hat{R} + \frac{1}{2} \hat{g}^{ab} \hat{g}^{ce} ( h_{ac,eb} - h_{ab,ec} - h_{ec,ab} + h_{eb,ac} ) + {\Delta^c}_{ce} {\hat{\Gamma}^{ea}}_a - {\hat{\Gamma}^{ec}}_c {\Delta_{ea}}^a - \hat{\Gamma}^{eca} \Delta_{cae} + \hat{\Gamma}^{eca} \Delta_{eac} $
$= \hat{R} + \hat{g}^{ab} \hat{g}^{cd} ( h_{ac,bd} - h_{ab,cd} ) + {\hat{\Gamma}^{da}}_a {\Delta^c}_{cd} - {\hat{\Gamma}^{dc}}_c {\Delta_{da}}^a - \hat{\Gamma}^{dca} \Delta_{cad} + \hat{\Gamma}^{dca} \Delta_{dac} $

Harmonic coordinate condition:
$\Gamma^a = 0$
${\Gamma^a}_{bc} g^{bc} = 0$
$\frac{1}{2} (g_{db,c} + g_{dc,b} - g_{bc,d}) g^{ad} g^{bc} = 0$
$(2 g_{db,c} - g_{bc,d}) g^{ad} g^{bc} = 0$
$2 g_{ab,c} g^{bc} = g_{bc,a} g^{bc}$
using weak field approximation, $g^{ab} \approx \hat{g}^{ab}:$
$2 g_{ab,c} \hat{g}^{bc} = g_{bc,a} \hat{g}^{bc}$
$2 (\hat{g}_{ab,c} + h_{ab,c}) \hat{g}^{bc} = (\hat{g}_{bc,a} + h_{bc,a}) \hat{g}^{bc}$

Harmonic gauge condition, weak field approximation, second derivative relation:
$2 (\hat{g}_{ab,cd} + h_{ab,cd}) \hat{g}^{bc} + 2 (\hat{g}_{ab,c} + h_{ab,c}) {\hat{g}^{bc}}_{,d} = (\hat{g}_{bc,ad} + h_{bc,ad}) \hat{g}^{bc} + (\hat{g}_{bc,a} + h_{bc,a}) {\hat{g}^{bc}}_{,d} $
$h_{ab,cd} \hat{g}^{bc} = \frac{1}{2} (\hat{g}_{bc,ad} + h_{bc,ad}) \hat{g}^{bc} + \frac{1}{2} (\hat{g}_{bc,a} + h_{bc,a}) {\hat{g}^{bc}}_{,d} - \hat{g}_{ab,cd} \hat{g}^{bc} - (\hat{g}_{ab,c} + h_{ab,c}) {\hat{g}^{bc}}_{,d} $
$h_{ab,cd} \hat{g}^{bc} = \frac{1}{2} (\hat{g}_{bc,ad} + h_{bc,ad}) \hat{g}^{bc} - \hat{g}_{ab,cd} \hat{g}^{bc} - \frac{1}{2} (\hat{g}_{bc,a} + h_{bc,a}) \hat{g}^{be} \hat{g}_{ef,d} \hat{g}^{fc} + (\hat{g}_{ab,c} + h_{ab,c}) \hat{g}^{be} \hat{g}_{ef,d} \hat{g}^{fc} $
$h_{ab,cd} \hat{g}^{bc} = \frac{1}{2} (\hat{g}_{bc,ad} + h_{bc,ad}) \hat{g}^{bc} - \hat{g}_{ab,cd} \hat{g}^{bc} + ( \hat{g}_{ab,c} - \frac{1}{2} \hat{g}_{bc,a} ) \hat{g}^{be} \hat{g}_{ef,d} \hat{g}^{fc} + ( h_{ab,c} - \frac{1}{2} h_{bc,a} ) \hat{g}^{be} \hat{g}_{ef,d} \hat{g}^{fc} $

Harmonic gauge condition if the background metric is cartesian then we get:
$2 h_{ab,c} \eta^{bc} = h_{bc,a} \eta^{bc}$
$2 {h_{ab}}^{,b} = {h^b}_{b,a}$

using expanded ${\hat{\Gamma}^a}_{bc}$ terms:
$R_{ab} = \frac{1}{2} \hat{g}^{ce} ( \hat{g}_{eb,ac} - \hat{g}_{ab,ec} - \hat{g}_{ec,ab} + \hat{g}_{ac,eb} + h_{ac,eb} - h_{ab,ec} - h_{ec,ab} + h_{eb,ac} ) - {\hat{\Gamma}_{ec}}^c {\hat{\Gamma}^e}_{ab} + {\hat{\Gamma}_{eb}}^c {\hat{\Gamma}^e}_{ac} + {\Delta^c}_{ce} {\hat{\Gamma}^e}_{ab} - {\Delta^c}_{be} {\hat{\Gamma}^e}_{ac} - {\hat{\Gamma}_{ec}}^c {\Delta^e}_{ab} + {\hat{\Gamma}_{eb}}^c {\Delta^e}_{ac} $
using harmonic coordinate condition:
$R_{ab} = - \frac{1}{2} \hat{g}^{cd} g_{ab,cd} - {\hat{\Gamma}_{ec}}^c {\hat{\Gamma}^e}_{ab} + {\hat{\Gamma}_{eb}}^c {\hat{\Gamma}^e}_{ac} + {\Delta^c}_{ce} {\hat{\Gamma}^e}_{ab} - {\Delta^c}_{be} {\hat{\Gamma}^e}_{ac} - {\hat{\Gamma}_{ec}}^c {\Delta^e}_{ab} + {\hat{\Gamma}_{eb}}^c {\Delta^e}_{ac} $

... but I should be looking for...
$h_{ab;cd} g^{cd} \approx g^{cd} \nabla_d (h_{ab,c} - {\Gamma^e}_{ac} h_{eb} - {\Gamma^e}_{bc} h_{ae})$
$= g^{cd} ( h_{ab,cd} - {\Gamma^e}_{ac,d} h_{eb} - {\Gamma^e}_{bc,d} h_{ae} - {\Gamma^e}_{ac} h_{eb,d} - {\Gamma^e}_{bc} h_{ae,d} - {\Gamma^f}_{ad} (h_{fb,c} - {\Gamma^e}_{fc} h_{eb} - {\Gamma^e}_{bc} h_{fe}) - {\Gamma^f}_{bd} (h_{af,c} - {\Gamma^e}_{ac} h_{ef} - {\Gamma^e}_{fc} h_{ae}) - {\Gamma^f}_{cd} (h_{ab,f} - {\Gamma^e}_{af} h_{eb} - {\Gamma^e}_{bf} h_{ae}) )$
$= g^{cd} ( h_{ab,cd} - {\Gamma^e}_{cd} h_{ab,e} - {\Gamma^e}_{bd} h_{ae,c} - {\Gamma^e}_{bc} h_{ae,d} - {\Gamma^e}_{ad} h_{be,c} - {\Gamma^e}_{ac} h_{be,d} - {\Gamma^e}_{bc,d} h_{ae} - {\Gamma^e}_{ac,d} h_{be} + {\Gamma^f}_{bd} {\Gamma^e}_{cf} h_{ae} + {\Gamma^f}_{cd} {\Gamma^e}_{bf} h_{ae} + {\Gamma^f}_{ad} {\Gamma^e}_{cf} h_{be} + {\Gamma^f}_{cd} {\Gamma^e}_{af} h_{be} + {\Gamma^f}_{ad} {\Gamma^e}_{bc} h_{ef} + {\Gamma^f}_{bd} {\Gamma^e}_{ac} h_{ef} )$
$\approx \hat{g}^{cd} h_{ab,cd} - 2 {\Gamma^{cd}}_b h_{ac,d} - 2 {\Gamma^{cd}}_a h_{bc,d} - {\Gamma^{cd}}_d h_{ab,c} + {\Gamma^e}_{cf} {\Gamma^{fc}}_b h_{ae} + {\Gamma^{fc}}_c {\Gamma^e}_{bf} h_{ae} - \hat{g}^{cd} {\Gamma^e}_{bc,d} h_{ae} + {\Gamma^e}_{cf} {\Gamma^{fc}}_a h_{be} + {\Gamma^{fc}}_c {\Gamma^e}_{fa} h_{be} - \hat{g}^{cd} {\Gamma^e}_{ac,d} h_{be} + 2 {\Gamma^{fc}}_a {\Gamma^e}_{cb} h_{ef} $

$R = g^{ab} R_{ab} \approx \hat{g}^{ab} R_{ab}$
$= \hat{g}^{ab} \hat{g}^{cd} h_{ab,cd} - 4 \Gamma^{bac} h_{ab,c} - {\Gamma^{cd}}_d \hat{g}^{ab} h_{ab,c} + 2 {\Gamma^e}_{ca} \Gamma^{acf} h_{ef} + 2 {\Gamma^e}_{ca} \Gamma^{fca} h_{ef} + 2 {\Gamma^{ac}}_c {\Gamma^{ef}}_a h_{ef} - 2 \hat{g}^{cd} {\Gamma^e}_{ac,d} {h^a}_e $

hmm...
if $g^{ab} \approx \hat{g}^{ab}$ then ${g^{ab}}_{,c} \approx {\hat{g}^{ab}}_{,c}$
then $-g^{ae} g_{ef,c} g^{fb} \approx -\hat{g}^{ae} \hat{g}_{ef,c} \hat{g}^{fb}$
then $-g^{ae} (\Gamma_{efc} + \Gamma_{fec}) g^{fb} \approx -\hat{g}^{ae} (\hat{\Gamma}_{efc} + \hat{\Gamma}_{fec}) \hat{g}^{fb}$