Well, I'm up early watching ryder cup and have to multitask while watching golf so I did a little searching. I make all my own cables. Here are my thoughts:
for low voltage DC the wire probably doesn't matter much. DC is much more friendly than AC since there is no inductive component. so you'll probably be fine with whatever you choose. I might go with 20/12 instead just because I could, assuming there is room to work with it on that crowded 12 pin connector you have to solder up.
quality of your workmanship is more important. It can be very hard to solder up a multipin connector like that without burning insulation off of adjacent pins. You can also accidentally unsolder adjacent pins while soldering up a pin. Make sure you start at the center and work your way out. I assume you will be reusing the old connectors so get a vaccuum desoldering tool (not expensive) and melt and suck the solder out so you can insert the wires into the pins making soldering quick and easy.
There are lots of good brands of wire out there, Belden, southwire are a few. I searched for 22/12 cable and found several options and one seller that sells generic cable by the foot. The alpha cable that the OEM uses is tinned and I like that but not mandatory. It is also more expensive so hopefully that is an indicator of quality. It is also shielded. Make sure you buy shielded and terminate the shield the way it is terminated on the original cable, probably terminated on one end only.
Mcmaster has several options of 20/12 cable shielded and they are very easy to order from and generally quality products but they don't have 22/12 shielded. If you think you can work with 20 awg wire, I'd probably just do that.