Run 15A 120VAC HeatPress on 15A outlet, limit current to 12A (limiter boxe???)
The easy way is to find a 24VAC transformer able to handle 12A at the output. Therefore 288VA AC or a little bit more. Connect the primary of the transformer in parallel with the 120V mains and the secondary side (24VAC) in series with the heating element. Beware to connect the secondary side of the transformer in such a way that the combined output voltage of the series circuit becomes 120VAC - 24VAC = 96VAC. This way your PID keeps the 120VAC and the heater can get 96VAC thereby reducing the current to the desired 12A.

You might try to keep the transformer outside the press and feed the same with only 96VAC. It is very well possible that the PID can work properly on 96VAC only