iPad randomly shuts off, can't backup

Over the past several weeks, my iPad has randomly begun shutting off, even when the battery is at 100% and plugged in to the original 12W charger. This initially happened once every few days, and has begun happening more often. Suddenly yesterday it got to the point where it won't stay on for even a couple minutes at a time. When it does shut off, it shows a dead battery screen.

I'm sure this is probably a battery issue, as it currently has 1117 cycles and is at 78% design capacity. What throws me off a bit though is that over half the time when I turn it back on, it shows me the multi-language hello screen and then says software update to 15.8.1 was successful. At first I thought this was a software issue, and it was running 15.7.9 and said as much. However after updating to 15.8.1 it still shows this screen when booting after a random shutdown.

I'm debating if it's even worth replacing the battery, given how old the device is. I've replaced the screen on this iPad before, not the most fun endeavor. I'm hesitant to buy any new Apple devices considering Apple's anti-repair practices, especially now that they're pairing iPhone batteries to logic boards.

In the meantime, I want to back it up so that I can migrate to a new device or not lose everything if I botch a battery swap. The problem is I can't even get it to stay on for long enough to do a backup. I've tried keeping it in airplane mode, and that hasn't helped. Is there anything I can do to get a backup out of it?