Here is a temperature plot from Cottonwood Lakes (elev 10,150) from May, 2022: CDEC Cottonwood Lakes
You also might want to check for a later date popping up on April 22 (7 AM PDT), as there could very well be a number of people not claiming their lottery reservations due to the heavy snow this year.
You probably won't need snow shoes, as there should be a well-beaten path through most of the route. But this year is probably the highest snow year ever recorded, so you will be hiking over snow, probably camping on snow even at Outpost camp. Once you climb the slope out of Mirror Lake (heading south, then turning westward), everyone will be off the main trail, as it will be completely buried. Forget the switchbacks above Trail Camp; just use "the chute" (the slope west of the switchbacks). You will need the ice axe to climb that area safely.
I am hoping the section past Trail Crest will be melted enough for you to travel safely.
Good luck -- this will likely be tougher than your winter fourteener in CO. Do not hesitate to turn back if it gets too treacherous.