It sounds like you do not have the correct configuration in Mach3. Check the ports and pins for output #1. It should be: Port 2 Pin 1 to fire the torch. If I remember correctly the plug was cut off to run your small Miller torch. The other problem may be that the pins on the plug side was wired wrong or that Hypertherm has changed the pinouts for the plug for firing the torch.
The only other thing may be that the THC option is off on the handset. If it worked on the Miller as you stated it sounds like the wiring is incorrect for the Hypertherm. Check with Hypertherm and make sure the pinouts are the same as the plug.