Check out https://www.nrcs.usda.gov/wps/portal/nr ... 2p2_046350
for both current and historical snow data
DO NOT plan to cross the border into the US. Big fines and jail.
Most hikers start at Harts or Rainy Pass, hike north and either turn around or continue to Manning Park with proper permit. Personally I like the Manning option especially if there is a lot of snow on the trail. Then travel (hitch, bus, trail angel) back to where you started and then continue south.
If there is a lot of snow on the trail when you start then start farther south thereby allowing more snow to melt-off. By the time you get to Manning you'll be ready for a zero.
As for your question regarding gear, I would consider traction devices and not snowshoes. In late June, even if there is snow on the trail it will be consolidated. There will be a few short sections where the trail is cut into a steep contour where post-holing could be possible but in MHO not worth the snowshoes. an Ice Axe for sure. Your feet will get wet for hours on end so be aware of frostbite (trench-foot). Adjust your shoes and socks accordingly.
Thumper aka dofdear