So, are you saying that Real Bout Fatal Fury Special did not work with the AR either?
If you have any, I would try a game that required the 4MB ram. Nevermind, I forgot that RBS is a 4MB RAM cart game--also forgot how damn good it is.
I haven't heard of the issue you're having, exactly, but it's a bit of a known issue with the AR that some games requiring the 1MB ram have various issues (eg. corrupted sprites in KoF '96, extra slowdown in Metal Slug, etc).
I have played Groove on Fight, and Real Bout Fatal Fury Special on a US console with an AR before, and I have also played them on a JP console with an official 4MB ram cart, both without any issues, but I would have to test your particular configuration to see if it works for me.
UPDATE: I tried Groove on Fight, and Real Bout Fatal Fury Special on a model 2 Japanese Saturn, with the Action Replay, and had no issues with either.