Thanks to all in the group for all the great advice and generous offers
to help. It's incredible how helpful the metalworking community has
been.

I have placed my order for conductive epoxy as I see this as the least
risky and least destructive means of getting a repair. I will also try
to use some silver/tin solder and see if this works better than the
lead based solder I've been using.

I can't tell what the posts are made of. Looks like a silver metal with
a gold plate.

In the long term I will also continue to slowly dremel away epoxy to
get to the post. When I get the replacement I'll upload a pic. I'm sure
that there is more post to work with since it broke off within the wall
of the speaker. Maybe if I'm careful and lucky with the dremel I'll be
able to expose enough threads to work with. Exposing enough post to cut
and then join ends together with a threaded sleeve sounds like a great
long term solution.

Any tips on what type of dremel head will cut epoxy but not damage the
post? Should I try to use any solvents when I get really close to the
post?