iPhone 8 battery at 1 percent and won't charge

My iPhone 8 (not Plus) went nuts yesterday. Its battery level dropped to 1 percent, and it won't charge.

It's a 256KB model, running iOS 11.3.1.

I've cleaned the Lightning port and tried a few different power sources. With any power source, the phone beeps when plugged in, but doesn't charge. Same with a Qi charging pad: beeps but doesn't charge.

I've tried restoring the phone using iTunes, but there isn't enough power, and the phone goes dark after a minute or so, then reboots. I've also tried restoring using the "hold down the power button when rebooting" trick. Same problem.

The phone is otherwise working fine, although when it was rebooting, it kept launching an app (the NPR News app, oddly enough!), even though I didn't tap its icon. I ended up deleting the app just to prevent that.

I'd be willing to restore it to my last backup (a couple of weeks old), but the phone doesn't have enough power to do so.

The Battery Health setting is displaying a dash alongside Maximum Capacity, and the Peak Performance setting says the phone can't determine battery health.

I "lease" this phone through the annual update program, so I know I can get a replacement. But I live a few hours from an Apple Store, and would prefer to just fix the one I have if possible!