I’d take a look at available networking solutions to see if any fit your requirements. I’d start here.
High level networking API’s for Unity typically have both the server and client running the same scene. That won’t work for you, so instead I’d look at low level options. If your other app isn’t even Unity, you may instead need to just use the Socket class.
Sorry I don’t know if VR has any specific networking requirements, and don’t know anything about bluetooth. Though I’ve seen some bluetooth assets on the Asset Store you could investigate. Networking should work fine without router hassle if you put the two devices on the same subnet.