I think I may have finally found a solution.
I found a 16 gallon trash can instead of the usual 13 gallon kitchen bin. It's 30" tall (our current one is about 22") and according to Amazon reviewers, is about 12 1/2" around at the base, so I'm going to put two 12" pavers inside. That should make it heavy enough to keep them from tipping it (20-30 lbs when empty), tall enough to discourage grazing, and still have enough room to use the regular 13 gallon bags!
I opted for the open top since the swing top will either get food stuck to the backside of the flap or it'll come off anyway when my two darling dolts get their heads or paws in it like
@Shezbug mentioned.
Thank you
@Destiny and
@Mizzely for the idea about adding weight to the bottom. I think that will be what finally does the trick for us.