Wiki has a little thing about print through that seems mostly right. It is an unstable thing, so rewind and 'rest' prior to playback can improve things a little. For critical masters, I've often wondered if interweaving paper or plastic leader tape between each program tape layer would help in the long run. The tape machine would have to look more like a flatbed film editor, though...
But for the apparent track alignment thing here, can you get some test tones recorded by Lab3? That'll help to A: set the PB azimuth correctly, using a scope and Lissajous figures. Then adjust PB height for maximum signal. Then re-verify azimuth. Then, B: Adjust record head so that PB head's output is the same as the Lab3 tape. This is all "walk it in, count turns of adjust screw, split the difference between dropoffs" stuff. If you can monitor PB while recording, this is easier. If you can monitor the record head as a playback head, then set it using the Lab3 tape, too. (Don't know if that's possible on the Akai.)
If you have a phase reversal switch available to you (via a mixer, maybe) then it's possible to get close without a scope. It's just easier with one.
Chip