Even better if you have a mains to 12v supply to plug it in to before you get to the car. Much better if your team mates don't open it up, rip open every bag in side and then leave them lying around before the 1st game, hence the reason both of mine have padlocks on the zippers!
Still the best way to shoot decent paint in the bleak midwinter, I have used them to good effect with Hellfire/Gold at this time of year.
As for the OP's battery Idea, work out the Ah rating of the battery and the draw of the bag and how much time a full charge will give you, if you think its worth it then go for it - if you feel its worth it.