What size is the broken tap, and is the item it is stuck in "portable" i.e. could you take it to a machine shop?

Just thinking that might be the best way of getting it removed, depending on size wondering it you could break it down using a small chisel and then drill a new hole and helicoil - just thinking out loud on that one, probably impractical.

Could try: https://www.2linc.com/product/broken-tap-removal-tools/#:~:text=Remove%20Broken%20Taps%20using%20Solid,Armor%20Mills%20and%20Armor%20Drills.

Re the badge hole, are you saying that running a tap through the holes wont allow the screws to freely go in?

Stupid question, are the screw threads on the screws in good nick, and the right thread?

As you know more often that not running a tap through a knackered thread restores normality unless the thread is completely knackered, or of course the tap is past its's best.