Yes, early June is late winter in Glacier. It may be hard to hike higher than the valleys. But you’re right that the Belly River may be sufficiently snow-free and you should be able to drive to that trailhead. It may also still be mostly snow-covered, too, but perhaps passable, especially if you bring some lightweight microspikes for traction.

You’ll find info on obtaining a permit and backpacking into that area in my e-guide to the CDT through Glacier—I started that trip from the trailhead you’re talking about. I think you’ll only be able to hike out-and-back from that trailhead but possibly reach some interior lakes for a couple nights of camping. That e-guide will provide a lot of detail for trip planning.

Good luck. Get in touch anytime.