There were very few genuine AGA games released, so the A500 would be the obvious choice for a gamer. An A1200 has lots of incompability issues with older games, albeit some can be fixed using degraders, WHDLoad and so on. If you want to use your 1200 for gaming you would most likely have to expand somewhat.
If you just want to break your joystick, use the A500. It's the far most compatible.