Ah derp sorry! I missed some key points in your first explanation about power and the current state of things. If you don't see the LEDs cycling when LSDJ is in master mode, but do see them on incoming MIDI events, then you know it has something to do with the game-link cable. I have included, as best as I could, information about how to figure out the wiring for the GameLink cable here. That is for a DMG-001 so your situation may be a tad different.
The cable I have is something similar to this one. I say similar because I ordered another cable from the same shop recently and it appears to be slightly different. Likewise, I tested it with both a DMG-001 and GameBoy Color (as my cable has connections for both).
I can't think of a reason why the GBA shouldn't work but I haven't tried it Also, LSDJ does work in Master Mode - it's just that I have not isolated the problem as to why I'm not getting MIDI Out events from the ShieldBoy. I wasn't planning on using that myself but do plan on figuring that out so folks that want to use the ShieldBoy have that option. I don't have an o-scope at the moment so it's a tad hard to track down currently.