How to Fix Busted Pipe Under Slab
TLDR: I ran brand new pipes down from the attic, and disconnected corroded old ones.
Update on what I ended up doing (the idea may help someone else):
I tried using plumber's epoxy on the pipe but it didn't hold. Then I tried a fiberglass resin, and that didn't hold.
Then, I called a plumber who basically said I'm SOL, but he offered to replumb my house for a little more than I'm willing to pay. But I liked his idea, so I disconnected all the connection points for that run of pipe, then dropped new PEX tubing down from the attic and tapped in at all those points. So now I have an unused pipe sitting in the ground, which isn't a problem for me, and I have hot water again.