Equation of Motion: Perturb Scalar Field and flat FLRW Metric (xAct)

Is there a way to get the equations of motion (eom) of a scalar field from a Lagrangian in a perturbed FLRW metric? I want to perturb the field as well as the metric to 1st order. What I did so far:

1. I derived the eom using xAct without specifying a metric.
2. I tried using xPand, but wasn’t convinced by my result:

$$
S=int!mathrm{d}^4xsqrt{-g}left(frac{M_P^2R}{2}-frac{g^{munu}partial_muphipartial_nuphi }{2}-frac{m^2(phi)phi^2}{2}-Vleft(phiright)right)+S_m.
$$

I gain the equations of motion using xAct and its "addon" xPand:

<< xAct`xPand`
$PrePrint = ScreenDollarIndices;
org[expr_] := NoScalar@Collect[ContractMetric[expr], $PerturbationParameter,ToCanonical]
collect[expr_] := NoScalar@Collect[expr, $PerturbationParameter, Identity]
order = 1;
DefManifold[M, 4, {α, β, γ, μ, ν, λ, σ}]
DefMetric[-1, g[-α, -β], CD, {";", "∇"}, PrintAs -> "!(*OverscriptBox[(g), (_)])"]
DefConformalMetric[g, a]
SetSlicing[g, n, h, cd, {"|", "[CapitalDifferentialD]"}, "FLFlat"]
DefMetricFields[g, dg, h]
DefTensor[sf[], M, PrintAs -> "ϕ"]
DefTensorPerturbation[pertsf[LI[order]], sf[], M, PrintAs -> "δϕ"]
DefScalarFunction[V]
DefScalarFunction[m]
DefConstantSymbol[massP, PrintAs -> "!(*SubscriptBox[(m), (p)])"]
L = Sqrt[-Detg[] ] (massP^2/2 RicciScalarCD[] - 
    1/2 CD[-β][sf[]] CD[β][sf[]] - V[sf[]] - 
    1/2 m[sf[]] m[sf[]] sf[] sf[])
varL = L // Perturbation // ExpandPerturbation // ContractMetric // 
  ToCanonical
0 == VarD[pertsf[LI[1]], CD][varL]/Sqrt[-Detg[]] /. 
   delta[-LI[1], LI[1]] -> 1 // 
  ToCanonical // ContractMetric(*Scalar EoM*)

Till here, everything seems to work fine. I get the eom:
$$
0=-m(phi)^2phi +nabla_anabla^aphi -m(phi)phi^2 m^prime(phi)-V^prime(phi)
$$
Now in the next step I get

eoms0c = Conformal[g, ga2][%] // ToCanonical

$$
0=-m(phi)^2phi +frac{nabla_anabla^aphi}{a^2} +2frac{nabla_aphinabla^aa}{a^3} -m(phi)phi^2 m^prime(phi)-V^prime(phi)
$$

But something is off here. For instance, I should have a $3dot{phi}dot{a}/a$ term, but instead here I have a $2dot{phi}dot{a}/a^3$ term.

3. I’m trying to use xCoba, but I couldn’t figure out how to do the perturbation of the field. Is there a way to use the metric-independent expression of xAct and re-evaluate it given a metric?