If you have real cart from the last generation of Pokemon, I would suggest transferring from the EZflash to the real cart. Obviously this would solve all your issues, but requires the use of either SP's or lower and a transfer cable. I can't foresee an easy way to solve that issue. The internal battery being dry shouldn't affect anything, I've traded with an actual cart that had that issue before.
Have you tried getting a different game (Ruby, Sapphire, Fire Red, Leaf Green), getting the box to 6 Pokemon, and seeing if you can transfer?