I’m a Computer Engineer and I have spent my whole career writing computer firmware, and believe me, you do not want the power to cut out when you are reprogramming the firmware. What you could end up with is some of the old SW mixed with some of the new SW and the chance of that working together is just about nil. In the best case you can power up the box and restart the update. In the worst case you could have a box that is dead and cannot be powered up. If you ended up with a dead 722, you could send it to Sound Devices and I’m sure they could get it reprogrammed and get it running again, but it would cost you $$.
If you have a known good battery that is charged, and if you HAVE to update the SW now, you could risk updating the SW with just the attached battery. However, your 722 must work just fine with the old SW, so why not wait to update for when you have internal and external power available?
And here is a dumb question: Why would you take your 722 anywhere without taking both an internal and external power source with you? Having redundant power sources is one of the great features of the 722, so why not use it?