Don't know if you've already seen this, but I found this website:
https://github.com/altr0n/arcade-classics-reBasically explains how to dump Basic Fun handheld arcades. Can't really provide any more insight past that but it might be useful (?)
I do know that the Qbert handheld hides some strange secrets - apparently it has Centipede (another basic fun handheld) coded inside, and I also found a test screen. Test screen is interesting as A. it doesn't appear on the centipede model, and B. it plays the same music found on all the recent bootleg handhelds (the badass song in the mogis and sy-88b among others). Maybe there's an entire menu in there, and they just forced some random pirated multicart to start at the qbert section?