That was basically the idea we had: dig a core across the back of the dam down to good subsoil, then backfill with clay up above the waterline. Essentially to try to build a new core in the dam.
I am finding right now the pond is filling slowly with no runoff going into the pond. Seems I an somewhat connected to ground water, but at a very low flow rate. I think my only real solution is to lower to the pond to a good groundwater level, or empty the pond, excavate it a bit, and backfill and seal with better clay than I have on-site. If I use clay on-site, I will need to clobber it with a sheepsfoot roller.
Re-work is probably never going to happen. Now I am forced to buy my wife some nice jewelry so I can stop hearing the complaining about how I wasted money on the attempt at the fix. ;-)