If it's starting in time, but gradually getting worse over time, then I
have nothing.
That happens when you sync to a loop which is too short or too long. Is it possible the metronomes are not the same in the two programs? Or perhaps one file or the other has a tempo change? I think I could start with a new autoload song, and open the tempo editor to make sure there is no tempo change. Set the tempo. Then import the audio and adjust the start anchor to start right on beat 1 bar 1. then open ultrabeat and put a kick drum with quarter notes, and loop it infinitely. Listen to see if stays on beat or drifts. If if it drifts, then select a section of audio - a long section which is long enouth to have the drift. Cut some number of bars - say 16 bars. Then choose 16 bars in the locators, and use the key command to adjust tempo, and see if it changes.
The only other thing I can think of is that Logic has a feature to make a audio play with a defined feel - a feature I never use, but that could mess up things if that were set.