First things first, you might want to look into making
reversible bumpers instead of two sets. Flipping a cloth panel is loads faster and more reliable than swapping entire bumpers. We actually missed a match at Lone Star thanks to a stripped riv-nut that made a bumper near impossible to remove.
We've never bothered trying to firmly attach the pool noodles to anything, if you do a good job pulling your cloth tight when you're stapling it down, then the noodles center up nicely and stay put just fine.
The GDC is pretty picky about the bumpers and what can go in them. I haven't seen anyone ask about adding Great Stuff to fill in bumpers, but I'm going to bet the reply would be along the lines of: "A Bumper consists of 3/4" plywood, two pool noodles, and a tough cloth covering, such as....". That's how they've answered question about having slightly less than spec bumpers. I can't see them looking kindly on bumpers that you're adding stuff to, even if it's just Great Stuff.
I like the idea of gaining strength from your bumpers. I'd look even harder at reversible bumpers in that case. Also remember that you have to pass inspection for size and weight without bumpers, so your bot needs to stay together without them.