Thanks for the reply :) Not running other programs, testing on 2 separate laptops, not using a hub, no SD cards inserted, HX Edit is the only program open.
I did some searching to find similar issues, and one user said that when he removed the HX Stomp driver everything started working for him (stable for days apparently). I just did that and so far, it's staying connected (fingers crossed - it's only been about 15 minutes, but that's a lot longer than before, at least).
So, if it's not a bug that's cool, but if it's caused by having other drivers installed like the Stomp, then hopefully Line 6 will take note as it could be an issue with the drivers conflicting.
I'll come back and update this if it disconnects again, but it seems promising so far.