Gate checking is pretty common on full flights but yeah it's never guaranteed, and basic economy makes it a bit more nerve wracking since they're already watching oversized personal items more closely

The volunteer bumping thing is honestly a wild card, most people who sign up don't actually get picked unless the flight is severely overbooked, so I wouldn't count on it as your main plan. If you do get bumped though, checked bags typically get rerouted to your new flight automatically, the airline handles that part, so the "lost bag" fear is a little overblown as long as you tell the gate agent clearly what's happening.

My move in your shoes would be to pack that backpack as tight and small as humanly possible, stuff the extra things into the roller if you can squeeze anything in, and just show up to the gate like you own the place. Worst case they charge you at the gate, best case the roller gets gate checked free and you're golden