I've made sound hole covers using foam poster board and card stock. Cut the foam board to a snug fit, and then glue it to a larger disc of card stock to keep it from falling in. This is a jury rig for experimental purposes, of course, but you could readily adapt that to nicer materials such as wood, and cork, or felt, for gaskets.
The lute rose type of feedback buster works by adding drag to the air flow through the hole at the 'main air' resonant pitch, which is the usual problem. Having a network of holes that cover the entire span of the hole, with a lot of edges, is the key to cutting down the amplitude of the mode without altering it's pitch much. A piece of window screen or gauze stretched over a ring would be extremely effective. The 'inverted wedding cake' parchment roses are the paradigm.
If there's a maker space near you it might be possible to get somebody there to make one up on a laser cutter, or you could join and do it yourself.
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