Drawing Power for Heating Pad

Heyo,

I'm trying to do similar project to the sparkfun Heating Pad Hand Warmer blanket, but instead of putting the heating pad on a blanket I want to put only one on a sleeve, also using a different micro-controller (Adafruit ItsyBitsy M0 express with the LiPo battery trinket that allows for JST battery connection). I also want my project to be portable so I'm using a 3.7V 2000mAh battery instead of a 5V power supply

Here's the wired connection guide I'm trying to follow:

My question is: how do I resolve the wire connections from the battery to the heating pads and MOSFET, but also have the battery connect to the board through the JST jack? I'm not exactly sure how they connect the battery to the heating pad joints. Would I have to cut the battery wire and solder my own new connections to it? Would this battery even be good enough to supply the heating pad with enough current?

Edit: the heating pad needs about 600mA at ~ 4V to get the desired heat output.

I was also considering having the board directly supply the voltage for the heating pad, but I don't think the board will be able to output the needed amount of current to heat the pad to the temperature I want.

Thanks!

2 Ah over 600 mA = 3 Hours, or 2 and a bit in practice.