So, a couple of questions:

* You mention the keyboard works fine with dongle. I assume this is a situation where the keyboard is using a Logitech uniFi dongle but the mouse is using Bluetooth?
* I assume this is happening even just in the shell (e.g. the Link outside of streaming mode)? Or is it only happening when streaming?
* If you use the keyboard to go to the Bluetooth menu on the Link when the mouse is disconnected, what does it show as? If it shows as disconnected, what happens if you try to connect it?

Beyond that, if you've enabled ssh on your Link, could you get /mnt/scratch/log/S99steam and /mnt/scratch/log/S13bluetooth and throw them somewhere you can link?

(If you haven't enabled ssh on your Link, you can do the 'Upload Logs' option in the system menu and then start a stream, which will copy a bunch of logs to your Steam host machine. But that gets a whole bunch of other extra data as well, where in this case just those two logs are useful.)