The way an actual GBA makes GB(C) games work is due to the GBA containing the hardware for both systems (which is why the Game Boy Micro didn't have GBC compatibility; it lacked the extra hardware). Effectively, it's two seperate systems in one.
GBA-enhanced games supposedly work by checking the hardware state on startup and registering this internally, then uses that to make things work differently. The game cannot and does not make any use of the added hardware; it doesn't have access to anything that isn't already on a GBC.
Hence, emulating GBA-enhanced functionality for a GBC-game would be done in a GBC emulator, not a GBA emulator, and it is there an option for GBA enhancements would need to exist.
While VBA-M, which Mednafen is based on, supports both systems, the core used in the RetroArch appears to only contain the GBA-related code, with Gambatte being used for GB/GBC support (as per SevenSidedDie's comment). The fact that Shantae, a GBC game, will not load on the Mednafen core, indicates that this is indeed true.
I am not aware of any emulators that allow these checks. Most games that perform this check do it to change the color palette to make the graphics look better on a GBA, in order to account for the different screens - but for an emulator, that's not normally something you'd want. You'd have to take that up with the emulator developers, though.
Best Answer
According to the Dolphin Emulator Wiki, there are no reported problems with the title. And, according to Bulbapedia, a Pokémon wiki, "Shiny Pokémon may rarely appear. They will always tip over when defeated. The chances of their appearance are unconfirmed, although they appear to occur more often than in the core series."
This seems to just be a case of really bad luck in your case. Unless your copy of the game is faulty, shinies should still appear, as Dolphin Emulator does not adjust any properties of your game.