What size screw is it?

Tough task trying to use a screw extractor on small stuff, as the hole you have to drill first (gives the extractor something to bite on) has to be smaller than the broken part of the screw yet big enough for the extractor to get started into.

I'd say just accept the misadventure as a learning experience, put another screw in about 1/4" away from the busted one. If you're planning to counterbore then plug screw heads, you may be able to dig enough of the epoxy & wood away to reach the embedded portion, maybe get a grip on it (visegrip's truly useful function!) then carefully back it out.

Oh, a tip: find some steel screws of the same size as your bronze ones. Drill proper pilot hole, run a steel screw in first to size each pilot hole, then remove. This eases the strain on bronze screws when you're running them into place.

(That last post was mine; I'd left it unattended before sending, my log-in must have timed out once I got back to it.)

Years ago I made an improvised plug cutter out of a piece of stainless steel brake line tubing with careful applications of a triangle-section file for cutting teeth in one end.

In a drill and using care you can use something like this to cut wood away from the broken screw portion enough to get a grip on it with needle-nose pliers. Leaves a rather large hole behind though so counter-bore then plug becomes worthwhile.  

There are now barrel and micro screw extractors   one of which should meet you nneed. After using either I would fill the hole will epoxy and wood flour and then drill a pilot hole after the mixture sets. Another option would be to epxy over the broken screw and continue to drill pilot holes and insert screws. 

   Not sure why the screw is embedded in epoxy, the thwarts should be removable for refinishing. If you can get the seat out hold a soldering iron on the screw to loosen the epoxy. Push or pull the screw out. Oh, the soldering iron should be hot to do this. 

George K