Do pilots have the ability to bypass customs when utilizing jump-seat privileges?
Just to put down as an answer what has been pretty well covered in the comments...

All pilots, jump-seating or not, will still clear customs.  They may have a separate line, but there will still be the normal record of their entry.

Beyond that, the airline absolutely knows who they let on the jumpseat for each flight -- so "no record" won't really happen in any event.