It can be done (actually, sounds like you have 3.3 months), but you’ll really have to cruise. Starting that late, you should be OK with the snow in the Sierra, unless there’s record snowpack this winter. The trail is generally graded for stock, so not usually too steep. But there are plenty of challenges along the way, it’s long, remote, dry at times, can get quite hot and quite cold, etc…
Another option to consider is to do just a “large section” of the trail. If you started at say Agua Dulce or Tehachapi on May 20, you’d meet more people, and have a more relaxing hike.
Jonathan