I remember getting caught in a snow squall late August in the Wind River Range :-)
You could just go and see what happens. There is no shame in taking a ride in a vehicle for a stretch if it is snowed in...or buying studded tires just for the crossing.
... and btw, no disrespect, but people from the west die in the White Mountains every year, mostly by underestimating the conditions we get above the tree line. Having said that, yeah, ... don't try to be a hero and watch the weather, be dressed for it and get a transport if your gear is not up to it even if it is for a very brief stretch