IMO, the best way to keep a
UGF in good shape is to put it on a shelf above the aquarium

. Bettas will do just fine without a filter as long as regular water changes are done. I'd probably change 1/2 gallon a week on a 1g tank. I'm not, as you may have noticed, a big fan of
UGF's. IME, reverse
UGF's work better and longer than the traditional type. I wonder, assuming you want to use it, if you could set it up reverse using one of the micro powerheads? This is where water is pumped down and flows up through the gravel bed. Bettas don't like current and you may not be able to do something like that in a 1g tank without creating too much current. The Aquarium Systems Mini Jets and Micro Jets are adjustable and might do the job with a little engineering.
Logan J