iPhone 5S stuck in bootloop with occasional red screen
I've been having issues with my 5S over the past few weeks. Started two few weeks back after I dropped it and the screen cracked slightly. A few days after, I had an issue where the phone stopped recognizing the SIM card (constantly said "Searching..."), but that was fixed with a software update. A few days after that, my phone started restarting intermittently, with erratic battery percentage readouts- sometimes it'd drop to low battery mode and need to be plugged in before it could turn back on, after which the percentage would jump back to ~60%. Never had it shutoff or get erratic percentages when on the charger, even if charging. The battery life has been poor for the past few months, so I chalked it up to early symptoms of battery failure and figured it'd be ok as long as I had a charger nearby.
Woke up this morning to the phone in a bootloop with the Apple logo. Turns on for ~10s, off for ~2s, repeat. I only get activity from the phone if it's plugged into an outlet. If I plug it into a computer, I don't see any activity from the phone or Win Explorer/iTunes unless I put it into DFU mode. Sometimes the phone will flash the red screen before shutting off, which I've read points toward prox sensor failure; sometimes it flashes the low battery screen before the Apple logo. Suspecting an issue with either of the two, but the intermittency of both makes me unsure.
I've tried swapping cables and a few hard resets with no avail, so I brought it to the local phone repair store. Their diagnostics found no hardware failure, and had no luck exiting the bootloop after power flushing (with the current battery and a replacement), and after testing with a new screen + digitizer.
At this point, I'm looking to see if I can get my 5S into a suitable state where I can back it up to iTunes again. Last backup was 10 days ago so it's not critical, but it would be nice to get that last bit of content. Not sure what steps to take next; hoping to avoid restoring/overwriting anything on the phone, and that maybe it's a corroded/faulty connection/component that can be fixed.