The main power need of the Square Stand is to charge the iPad. There are multiple sources you can use for power to run it. I have not used any of these specifically but we do use Anker power batteries all the time for portable charging.
If you have the room you could also bypass the fanciness of the compact lithium ion battery packs above and go old school large car battery plus a power inverter I have this one from amazon, power inverter. That should get you a lot longer battery life that the compact one but obviously it is not compact and would take a car charger to charge the battery overnight rather than just plugging it in.
