Hi, Mia! Welcome to HST!
I can't answer your snow question with precision, but I imagine it will be getting fairly clear by then.
I will offer a couple of possible things to include on your trip, though. From Dicks to Fontanilis is short and there is a beautiful long flow over the granite pluton out of Fontanilis. You can follow the stream down the granite then XC a very short distance back to the trail.
A day hike climb of Mt. Tallac from Gilmore offers an astounding view of all of Tahoe from 3000' directly above the lake. Gilmore is a good base camp for that climb, but shortly after the snow is gone the grasslands around Gilmore can be a mosquito haven, so be prepared.
There is a very good out of the way camp site on Susie Lake. Cross the stream at the outlet, continue on the trail, then when the trail turns away from the lake, follow the edge of the lake several hundred feet until you come to a little peninsula with a couple of nice flat camp spots. There is good fishing in Susie.
If there is still snow covering the trail just above Heather, be careful as it can be steep and slippery. Ditto if there is still snow on the trail along the side of Heather.
Here is a map that covers much of your intended route.
Have great fun and be sure to post a detailed trip report with lots of pictures when you get back!!
PS. If you ever decide to do a longer version of this route, you can continue on to Genevive, XC due west (possibly using the previous trail which is no longer maintained) to the General Creek trail, which then runs into the PCT.
This map will give you some ideas.