Hmm, was unaware of so much as a proof of concept of a megadrive emulator for the GBA (homebrew on the GBA is a thing I cared a lot about, the megadrive almost as much. Still know of none. It is very rare for repro type people to make their own emulators too). Never mind something vaguely playable like that and even attempting sound. Wouldn't mind seeing that get opened up a bit.
If the basic dumping software does not handle it then you have to go custom. Few have ever bothered to investigate repros/multicarts at this level -- two main choices are either a la NOR flash carts (they have commands to send the cart to basically) or something similar to
https://mgba.io/2015/10/20/dumping-the-undumped/ with the sensible money going on the former. Old school flash carts did something called page switching -- you put your game on a page of NOR, boot into a loader, have the loader set the relevant page of NOR as the primary and then reboot (soft boot looking at that video as there was no GBA logo again).
2013 vintage repros had not quite gone for the blob of epoxy approach seen with current ones. Might then be a fire card or 3 in 1, more likely the former. However if they were good enough to make an emulator then all bets are off.
That menu looks a bit reminiscent of some types of flash cart loader, but centred white text on a black background does all blend together after a while so don't go in too hard for that one.
Figuring out what to do for old school NOR style cart is tricky. Two main approaches are take the case off and see if they left chip markings. Said chip markings will hopefully lead to spec sheets and then onto what commands need to be sent. Option 2 is a something has to be there so what it will mostly likely present the GBA by default is the loader. Said loader should then be able to be figured out via emulation what commands to send, at that point you would probably want either a few wires inside the cart to send the relevant command or to rewrite the dumper (that one for the shrek videos above probably being a good base).