What alternative methods exist to correct a belly in a clay sewer line?
The best option that I can think of would be to have the sewer lined.  This company is simply the first Google search result I found:
http://www.dontdig.com/drainlining.php
The premise here is that a flexible liner is inserted into the pipe (usually using air - think of it as turning a sock inside out) and then the liner is hardened in place (water activated? resin?). The new liner will bridge any cracks or breaks in the pipe as well as any gaps in the joints.
I would also suggest actually getting the section of drain in question inspected...you will really want to know what the true problem is before deciding on the proper corrective action.