2014 Prius C: Having trouble starting

Hey everyone, I'm hoping you can help since so far the dealership seems even more confused than I am.

A few weeks ago my 2014 Prius c (55,000 miles) started not starting until multiple attempts. The car unlocks electronically, the lights come on (although dim) and when I turn the key in the ignition, first the check engine light comes on and nothing else happens. The next few times I turn the key more lights on the dashboard start lighting up, but the car never fully starts up. Then eventually, after 10-15 tries (with various attempts to reset like getting out of the car and locking and unlocking it, messing with the headlights, shifting) the car starts, and drives fine, and even starts again 2-3 hours later with no issues. Then, after letting it sit for a few days, the next time I try to start it the same thing happens again. Today, the car dashboard came up with a lot of the warning lights still on, but every time I pressed the breaks or tried to shift it seemed like it rebooted: the screen went blank, then all the lights came back up again. The weird thing that happens during this is that the screen displays the stats (distance, time and MPG) from the last time it was driven, like it thinks it never turned off.

I took it to the dealer today (needed 55,000 mile service anyway) and first they didn't believe me (from the notes they wrote it looked like they thought I didn't know how to shift the car), then when I tried to take the car it wouldn't start properly and they seemed very confused and worried. The guy mentioned something about how the cover on the shift lock override button was missing, but I'm pretty sure that's been missing for a while now and these problems are quite recent.

They also originally tried to get me to buy a new battery, but the last service I had last summer they told me the battery was fine. And then when I had the guy get in my car and see was was actually happening he seemed convinced it wasn't the battery. I do park outside (but it's been such a mild winter) and drive ~2 times a week for ~20 minutes at a time, so I wouldn't be shocked if it's the battery, but I'm skeptical, and I don't want to buy a new battery if mine is fine. They checked it today and said it was at 50% charge.

I'd love insight into what it might be, or things I can do to diagnose further! I'm worried it's a software issue, and that I'll spend a bunch of money because the mechanics keep diagnosing mechanical issues, but no dash lights are on, so I'm not sure how to diagnose that.