Re: EASY problem
The battery you are using is the wrong power source for the wire; or does not have the watts-hour capability; needs to chemically recover itself for a few minutes after delivering all it can deliver for a few seconds.
A better option could be a less voltage power supply and capable of more current, or Watts. Something like 1 volt at 20 Amperes by guessing. The exact values are to be tried by reaching the needed temperature of the wire and not more.
Measure voltage, current and temperature and make a chart.
If you can only use 12V, connect a 12V light bulb in series and check it the temperature reaches your needs to cut foam. The thickness and lenght of the hot wire has a lot to do with the success. If it is from a guitar, meaning a couple of feet, you must lower the voltage and increase the current capability of the power supply.