In France, an extremely gifted, aging, and seriously ill writer, supported by a female student, starts his last desperate battle against the critics and his best friend to achieve the global recognition.

Thanks.

The writer (my protaganist), 65, suffers from asthma and heart issues - I would say:
he's at the treshold of death.

Both his best friend and the critics are against him.
I don't think I have to reveal their motivation in a logline. Please compare:

"A former super-actor battles his ego, family, and cast while putting play intended
to revive his career." ("Birdman").

Finally my new version (V3):

France 1965. A writer, at the treshold of death, battles his best friend and critics to win the Nobel Prize.

Thanks again.

When I read aging, I assume the writer is in his mid to late 70s. I think it would be better to mention the age since you also go on to state the writer is seriously ill. Why is (whatever he's working on) his last battle against the critics? Is he terminally ill and dying? If so, then you could also mention that it gives your logline a ticking timebomb. And who's the main antagonist here? Is it the critics or his best friend? What did his best friend do that the writer is so against him?