Any smart lock should work on an interior door, you may have to drill a deadbolt but its the simplest solution. Switchbot and U-tec both make smart locks with fingerprints and excellent Home Assistant integration. Ubiquiti also sells magnetic latches you could install for access control if you're in their ecosystem.

If you can give up on Home Assistant integration/smart features, however, there are oodles of push-button and even fingerprint door (without other connectivity features) knobs out there. Most of them are terrible quality and not secure but that doesn't really matter to you and they are cheap.

Are these miscreants two-legged or four-legged?

Sliding old school deadbolt at the top

I like your first idea better. Why does your wife hate fun?

They make maglocks that you could put in the corner of the doorframe. Then wire it to an esp32. Esphome to connect it and home assistant to control.

I use one for a liquor cupboard that auto locks if my wife and I both leave the house. (well our phones anyway)

To be real...kind of all of them. All the "smart" locks you see for sale are WILDLY insecure at the consumer level.

You are correct that almost all of them focus on the deadbolt, but look at something like this. I kind of hate this company, but they make components as well as entire lock systems.

I have no idea if they work with HA or not, just giving you a jumping off point to familiarize yourself with the options and terminology to have a deeper search.