In reality, it is the pilot, the instructor and the DE who must be
satisfied with the logging. Most instructors and most DE's are onboard
with both pilots logging PIC under this circumstance (but not all). Not
sure enough instructors or DE's have seen this approach to logging
cross country to know what they think. Also if you are headed for pro
pilot with a major airline, your logs will undergo scrutiny. Will they
be happy? Who knows. If you, your co-pilot, your instructor and your DE
are happy with it, go ahead and log it, if you really need the cross
country time. But realize you are logging something that is a bit of a
gray area and it may come back to bite you. Then again, maybe not.

Whatever you do, make sure you have the skills to plan a long cross
country flight through timezone changes, prevailing wind direction
issues, length of day issues, and weather issues to understand them and
be able to anticipate their effect on your flight. Also make sure you
can plan your flight into an airport you have never been to before and
all that entails. as well as deal with complicated airspace. If you can
accomplish this with your method and everyone will sign things off, you
may be ok.

No reason why BOTH of you cannot plan the flight seperately and compare
your plans and choose the best of both. Could be interesting and a good
learning experience.