the Coastal Mountains, esp. north of Vancouver, or the Rocky Mountains around Banff, Jasper, and the Purcell's?
I would suggest a loop. I'll describe it one way, but it could be reversed.
Starting in Montana (I15), AB22 to 40 into Kananaskis country. Check the opening time for Highwood Pass on 40 (mid June?). These AB prov parks rival Banff etc.
Canmore (there was major flooding in this area in Mid June several years back), Banff, Lake Louise, Jasper. Possible side trips into Kootenay and Yoho.
Mt Robeson (16), 5 S to Kamloops, with side trip into Wells Gray (lots of waterfalls, lakes)
1 to Cache Creek, 99 to Lillooet, and S to Whistler. Best access to the Coast Mtns. Be prepared for grades. I spent several weeks make a couple loops on backroads in this area.
Speaking of Backroads - check out the Musio Backroads Mapbooks (like Delourme Atlas, only better).
99 to Vancouver, 1 and 3 east. Manning Prov Park. Nelson area, Kootenays. If there wasn't time to go all the way to the coast, I'd consider making a shorter loop from Kamloops to Nelson and 3.
Return to the USA on 93/95 or via Glacier (the US one).
The main thing that misses the Hwy 1 through Golden, Glacier NP (CA) to Revelstoke.
By July you might have to adjust your route around forest fires. I think schools are out late June. July 2 is Canada day, so the long weekend will be as busy as in the USA. I'd plan reservations, or settling in someplace with hiking access before that.
Browse the BC Prov parks site.