Does crystal field splitting energy Δ increase going down a group of transition metals if the ligands are weak field?
I see where you are coming from, but you forgot that even with π-donor ligands, there is still a σ-bonding present in the complex. In the 4d/5d metals, the $\mathrm{e_g}^*$ orbital is pushed up by a lot due to the σ-type overlap.

The $\mathrm{e_g}$ overlap is always stronger than $\mathrm{t_{2g}}$ overlap because the metal $\mathrm{e_g}$ orbitals point directly at the ligands ($\mathrm{d}_{x^2-y^2}$ and $\mathrm{d}_{z^2}$), whereas $\mathrm{t_{2g}}$ orbitals point in between the ligands ($\mathrm{d}_{xy}$, $\mathrm{d}_{xz}$, $\mathrm{d}_{yz}$).

So, yes, the $\mathrm{t_{2g}^*}$ orbital is pushed up higher by the π-donor, but the $\mathrm{e_g^*}$ orbital is pushed up even higher. Thus, for example, the complex $\ce{[RuCl6]^2-}$ has a $(\mathrm{t_{2g}})^4$ configuration (i.e. low spin) even though $\ce{Cl-}$ is a π-donor ligand.