When it comes to polynomial Szemeredi, there are no better bounds known for coloring than for density, except in cases where the only bounds for density are from Bergelson-Leibman and the bounds for the coloring versions are then just from Shelah, Walters, etc.

Great! I think I do need something independent of d to actually get primitive recursive bounds in my paper. It would be nice if someone could fix the "problematic" proof...

I just chose it because it is a function that grows to infinity very slowly, so that $|A|=o(p)$ but $A$ would still be pretty large.

Good catch, although your comment only applies to the squares case. For squares I think $L(p)$ is bigger, because for instance we can use the squares of everything up to $d\sqrt{p}$ and then we expect a $2\sqrt{p(1-d^2)}$ increasing sequence left that is compatible with this. This can yield anything up to $\sqrt{5p}$ although empirical evidence suggests $\sqrt{6p}$ is the answer. For inverses, cubes, fourth powers, and so on, empirical evidence suggests it appears to approach $2\sqrt{p}$. I am pretty sure that squares are the only exceptional case.