I think the 1200 would probably be about the worst choice for a BBS (at least if you keep it in the desktop case). They seem to have the most heat issues of any Amiga. 2000s and 3000s are good. If you don't need much expansion, an A500 would be a workhorse, as well. (If you DO need something more than the 7mhz 68000, the A500's desktop case can start holding some heat, much like the A1200.)
The biggest problem with using any Amiga as a BBS is getting reasonable I/O speeds out of it. The built-in serial ports were quite painful, even back in the day. I had to run a multi-serial board just to get reasonably stable 38k connections... Of course, a much better solution nowadays is an ethernet card and a bunch of virtual connections.
