Coordinate transformation for perturbed stress-energy tensor

I am reading Baunmann’s notes on cosmology and am having difficulty with the exercise on pg. 75.

Consider the perturbed stress-energy tensor stress-energy tensor
begin{align}
T^0{}_0 &= bar{rho}(eta) + deltarho[0.25cm]
T^i{}_0 &= [bar{rho}(eta) + bar{P}(eta)]v^i[0.25cm]
T^i{}_j &= – [bar{P}(eta) + delta P]delta_j^i – Pi^i{}_j
end{align}

Under the change of coordinates,
begin{equation}
X^muto tilde{X}^mu equiv X^mu + xi^mu(eta,{bf x}),
quad
{rm where}
quad
xi^0equiv T
,
xi^i = L^i
end{equation}
we know
begin{equation}
T^mu{}_nu(X)
=
frac{partial X^mu}{partial tilde{X}^alpha}
frac{partial tilde{X}^beta}{partial X^nu}
tilde{T}^{alpha}{}_beta
.
end{equation}

Baumann then states that this gives,
begin{equation}
begin{aligned}
delta rho & mapsto delta rho-T bar{rho}^{prime}[0.25cm]
delta P & mapsto delta P-T bar{P}^{prime}[0.25cm]
q_{i} & mapsto q_{i}+(bar{rho}+bar{P}) L_{i}^{prime}[0.25cm]
v_{i} & mapsto v_{i}+L_{i}^{prime}[0.25cm]
Pi_{i j} & mapsto Pi_{i j}.
end{aligned}
end{equation}

I am having difficulty reproducing any of these results. Note that,
begin{equation}
frac{partial tilde{X}^mu}{partial X^nu} = delta_nu^mu + partial_nu xi^mu(eta,{bf x})
.
end{equation}

If we assume that $xi$ is infinitesimal then to first order in $xi$,
begin{equation}
frac{partial X^mu}{partial tilde{X}^nu} = delta_nu^mu – partial_nu xi^mu(eta,{bf x})
.
end{equation}

I’ve tried substituting this into the tensor transformation law and comparing the relevant coefficients, but this doesn’t seem to work. I’ve also tried inverting the tensor transformation law to calculate the elements of $tilde{T}^{alpha}{}_beta$ explicitly but this doesn’t seem like the right direction either.

Any help would be appreciated.