3) Plot out a few distributions for key sites: dry midlatitudes, wet midlatitudes, polar, tropics (monsoon-dominated). If these PDFs look pretty similar, you might be able to figure out a sensible approach.
I think the key difficulty here isn't going to be that different places have different total rainfall, but that different places can have rainy seasons of different lengths. If the season was always 30 days, I think this approach would be fine. I'm really interested in seeing those CDFs just to see how different they are.
Let us know what you come up with!
Statistical tests for extreme precipitation volumes by V. Yu. Korolev1, A. K. Gorshenin2 , K. P. Belyaev3, 29 Nov 2018