I am not sure, but it could be that the sample rate you are using is not implemented into OMF, it's a bit of an old standard, OMFI is only 16 bit, but OMFII is 24. Do you export an OMF with embedded audio, maybe it will work when you only reference the audio. I've done an export from Nuendo to DP this way, which worked allright.
If it doesn't work then I would just consolidate each track into a long file, maybe with every file starting at the same time, it's not necessary with the embedded time stamp but it will make placing audio a bit easier and you can absolutelely be sure that all tracks are in sync.
I thought Logic had some sort of "bounce all tracks" export feature for cross-platform operation?
Contact the other engineer, probably he doesn't mind long files.