I have to admit, this will be a little more to do, then I’d wished for you… 
OK, first thing, the included/built-in BT adapter is not a good idea to work with. They tend to be unreliable, mostly because there is to much “noise” around them. If you want to go with a BT adapter on the same machine, I’d strongly advice to get a dongle and a USB extension cord, so you can put the BT dongle at least 15cm away from the NUC.
But there are other ways around, that’s why I was asking about the distances. I’ll give you some alternatives, which one fits best for your living space is up to you. 
No particular order, just as they came to mind… 
One method to track BT devices is a tablet or old cellphone, where you install the HA companion app on. The app can monitor BT tracker and report this back to HA.
You can setup a BT proxy with an ESPHome configured device, that you put on your hallway cabinet. This ESP32 would take “everything BT” and just sends it to HA. Imagine it like you would have a really long extension cord for the BT dongle. Handling of data is on the HA server.
You can setup the ESP32 as “BT dongle” itself, so the data would be processed on that ESP, and is then send to HA. Data handling is on the ESP.
If you have an old Pi or other computer lying around, depending on the model, they do have BT on board (here goes the same as for the NUC, an extension cord is practically mandatory).
Let me know, where you want more information about, so we can figure out how to solve this for you.