I got it with Alloy. I found the best way to go was by standing on the top platform until they were about halfway through jumping at me, and when they did that I'd jump to the other platform away from them. When they jumped toward me, I'd jump back to the upper platform, passing harmlessly below them. This gave me the greatest chance of getting any powerups, and I'd only attack with invulnerability or the gun.
Also, beware of pausing! I paused the game to take a phone call, and after having done so, the range of the green orbitals was about 4x as big for my opponent, but still the same size for me. Almost all of the life I lost from Bebe (when I took the call) on was from getting hit by those green orbitals when my opponent was jumping from the upper platform, and I was dropping down to the one below; literally the entire height of the screen between us. If not for that, I'm confident I would have finished all 12 enemies instead of dying on Convict.
At any rate, another game in the bag. I'm definitely coming back to this to get my 12 kills in at some point, but now it's dinner time.
Thanks for the help everyone, choosing Alloy was probably the best move I could have made.