Bit needed--but does it even exist?

Submitted to Product Talk

I'm restoring an old wood table that not only sat in a basement for 30+ years and had things piled haphazardly on it, but has been through two storm surges from hurricanes.

The table was assembled with pegs, which were in turn covered by inset wood caps. Some of the caps are missing and some were loose enough to pry out with my fingernails; I'll replace the missing ones with doweling cut to fit and re-glue the loose ones.

Though the table itself cleaned up well, the wood underneath the removed caps isn't in such great shape: it's not quite rotted, but I think it should be removed. I was thinking that my Dremel (an 8220) coupled with the plunge router accessory would work, but I'd like a bit that removes material downward only and not to the sides--i.e., that I could set a depth and it would evenly clean out the soft wood below without (accidentally) making the hole wider. But I'm not seeing any such bit in the Dremel accessories, and was wondering if there was any third-party bit that would do this.