Software issues and problems:
MACH3 isn't designed for DIY CNC Wirecutters.True enough. If you're not too invested in Mach3 you should try out EMC2. I know for a fact that 6-axis wire EDM's have been successfully built with EMC2 as the controller.Another possibility is perhaps there are other PC CNC controller software that has this function REVERSE RUN, PAUSE, RUN CYCLE.
Secondarily, the HAL in EMC2 should allow you to bypass the issues you're having with Mach3 cycles and 'buttons'.
The learning curve for EMC2 is a bit steeper than with Mach3. A familiarity with Linux, ladder logic/PLC programming and hardware interfacing is helpful if not necessary. Considering the content of your post, you shouldn't have a problem. There are lots of resources available on the Internet covering all aspects of EMC2 and Linux in general.
A link to the EMC2 project: LinuxCNC.Org