Removing old grommets

I know this isn’t a building question, but this isn’t a simple maintenance question either. I picked up a McSpadden made in the early 1980s and the tuners won’t stay tightened, even when I adjust the screw. I picked up a set of planetary tuners when I was in Mountain View and I’m attempting to install them myself. I know I’ll need to enlarge the hole a bit (I have a reamer) but I can’t get the old grommet out of the dulcimer. I’m afraid to just pull it out. Any suggestions? TIA