Would it be sufficient here to write Ne4+ (meaning Nce4) only (i.e.
without the "c" but with "+"), as Nfe4 would not be check?
It depends on exactly what you mean by "sufficient". By "normal" definitions of "sufficient" the answer is obviously "yes". "Ne4+" disambiguates and leaves the human reader in no doubt which knight was moved.
However it doesn't work in FEN diagrams. You need to specify Nce4 to get it to work. So, then the answer would be "No".
The FIDE Laws of Chess describe how you should annotate your games in Appendix C and that does not allow for using check as a way of disambiguating between two different pieces of the same type moving to the same square. However there is no penalty specified for mistakes. It is up to the arbiter to decide and that will only happen if it is noticed.
In real life the most that might happen is that if the games are transcribed for sending to FIDE or for publishing the player might get a scolding from the arbiter tasked with this chore, but this is extremely unlikely in this case and would happen long after the game is completed.
Mistakes far worse than this happen all the time and the only comeback from the arbiters is when the scoresheet is completely illegible or grossly incomplete. Vigilant arbiters will notice these misdemeanors during the game and warn the player concerned to mend their ways.