Nice!
If I understand correctly, you have a positively constrained variable and you want to allow values near zero (which is where it reduces to generalized gamma). For this, about the only choice you can make is something like a half-normal or half-t or exponential. The difference is that half-normal/half-t bounds the density at zero, whereas with something like exponential(1), the density is unbounded at zero. After that, it really just depends what you think large values might look like to be consistent to give you a sense of what scale to use.
As always, I’d recommend simulating and see how it works in cases where you know the answer. It can be very numerically unstable to try to push a positive-constrained value to zero, as it entails pushing the log of the value to negative infinity. If you try to put a prior on P_log, you won’t be able to make it consistent with zero and proper.