For awhile I was having a lot of trouble with apps or certain games not showing on the monitor I wanted but I started using Borderless Gaming app as I was also having trouble with a few games going fullscreen but it has an option to specify which monitor you prefer for any game or app you could try it. Just have Big Box running and use the right arrow to put it on the right side and right click on it there and choose the monitor you want. Run Borderless Gaming with startup and it should always do it automatically. It also should help with Big Box not being full screen.
I'm not saying you're doing anything wrong but why not just use Duckstation in Launchbox with fullscreen? Then you wouldn't need bezel support. Because for that I use
$Esc::
Send !{F4}
Process,Close,DuckStation
Run,taskkill WinTitle /im "duckstation-qt-x64-ReleaseLTCG.exe" /F
*take note of your Duckstation version .exe and replace it if different. They've used different .exe names with different versions.
Or you could replace Duckstation with RocketLauncher, but I don't know if it will work or not.
$Esc::
Send !{F4}
Process,Close,RocketLauncher
Run,taskkill WinTitle /im "RocketLauncher.exe" /F
Sometimes different emulators require different scripts for them to be closed.
If you prefer bezel over fullscreen no problem, I just prefer all my games fullscreen and am more inquiring if you need help with making Duckstation fullscreen but if you want the bezel over fullscreen I'm not trying to change your mind.