Apple will replace the battery for around $50. Though their stores are closed so you might have to send it in.

Another option is to get a battery replacement kit from iFixit.[1] I did that and it doubled my SE's battery life.

1. https://www.ifixit.com/Store/iPhone/iPhone-SE-Replacement-Ba...

Have you checked the battery health/cycles? Mine was like that, even iOS at once unexpectedly shutdown...

The battery page in settings, under Battery Health should tell your capacity and other info. for me a new battery fixed everything... and batteries aren't usually covered under warranty as far as I know in general but Apple replaced it for me. FedEx Overnighted it to their repair center in California, and then back. So only without a phone for a few days. Kinda like a brand new phone just by changing the battery.

That's the thing. The battery is still healthy and it looks like it retains 90% of its capacity on the Battery Health setting page. I bought my SE very late into its lifecycle (June 2018), so I'm not sure why it's acting this way. I have had a problem with the phone rapidly losing charge in cold weather though. Did you also have severe battery drain on your SE on iOS 13 before replacing the battery?

Yeah, at the point before I got it replaced seemed like sometimes you could watch the percent going down in real time. But then at times I’d be at a few percent and it’d run lot longer then I expected.

I know the OS itself prompted me to get a repair. Got a popup when restarting about a unexpected shutdown. So maybe that helped with the replacement process.