I'd go for removing the CPU card since these problems you're describing sounds a lot like CPU or memory issues to me.
If it works without CPU card then next is replacing the simm since it's the cheapest option. Also if the 060 doesn't have any cooling option (heatsing or better yet heatsink with fan) then add one cause 1260/60 are usually overclocked 50mhz models and they do get hot. Next would be to try the card in a different not expanded A1200 to see if it might be power/compability error and if that's not the case go for yet another different simm (again cheaper option) and if still nogo replace the CPU.
If removing the CPU card doesn't do it for you then I'm a bit lost... I'd go for removing as much addons as possible (even HDD). Boot of a floppy and see if it still crashes after a set period of time.
