I like that the first idea for a non-destructive technique was shooting fire at wood.
When I said butane torch, I meant a dinky little Bernzomatic micro torch - basically, a glorified bic lighter. It's not even all that focused/powerful of a flame. Was thinking of holding it at a distance underneath the brass badge - so, coaxing warmth towards brass, not 'shooting fire at wood.'
Turns out the (waxed) floss suggestion worked like a charm for the removal - took about 10 minutes of pretty forceful 'sawing' through the adhesive starting from the one corner out of four that I could get under, but that was definitely best case scenario for removal. Thanks, bainjs.
Some of the adhesive stayed on the back of the badge, too, so I'm optimistic that I can either re-activate what's there with some light heat or otherwise use some super-light sanding there & on the mounting point with a drop or two of the wood glue for a new adhesive.
Thanks to everyone that chimed in.