This is the exact same issue I've been struggling with since buying a Big Box license. I wanted to use LaunchBox together with Steam Link so I could play emulated games on my TV via Steam Link. I can start BB just fine, and load any game, which plays wonderfully. The issue is that when I exit the game, I see the BB interface for a few seconds, and then I'm kicked back into Steam's Big Picture Mode, which doesn't seem to think that BB is still running.
The only way to fix it is to run up a couple flights of stairs to my computer, alt-tab to BB, close it, and then open it again with the Steam Link.
Has anyone else run into this? I haven't been able to find any solution.