First off, I have no robot building experience, so take the following with as many grains of salt as needed.
You mentioned that increased traction would lead to parts breaking in a collision. Have you done any experimentation with other filaments, like Cheetah or Armadillo? Both are on the "flexible" spectrum, but both are fairly rigid and print in most stock printers. Armadillo in particular is designed to be tough -- it's really only flexible so that it gives instead of breaking. They compare Armadillo to hard hat material. These filaments do cost more than PLA, but if one "unbreakable" (yah, I know, no such thing) assembly in Armadillo replaces 2-3 spare PLA assemblies, the economics might work out in your favor (plus fewer breakdowns in competition).
I was initially thinking that something like NinjaFlex might be useful for the wheels for traction, but honestly, NinjaFlex is pretty slick (has to be, to slide through all the tubes and things necessary to print it), so there's probably no gain to be had on that front. I made a NinjaFlex glove for halloween (HellBoy) and it was really hard to hold things because the surface was as slick as PLA.
As a side note: I know that Makers Muse used PC Max polycarbonate for some gears in one of his battle bots and was very happy. I have absolutely no experience with polycarbonate printing, myself, though I do know that it requires a somewhat special printer to pull it off...
(No, I don't have any association with Fenner Drives / NinjaFlex. But I have been impressed with how tough their material is. Makes sense, given that they made custom drive belts before they got into filament work, so they have some material engineering experience.)