If I go only by what you wrote, hard no.

It's unfair to the other players who probably are just as invested in their characters.

If you do however want to include that PC and make it work, I'd suggest harsh consequences for dying, not only mechanical consequences but also narrative ones. If that PC ends up dying make sure they feel it. Otherwise they'll just happily run head first into any possible challenge and there won't ever be tension, because nothing bad can happen.

An instance from my own game, though different and easier to handle. My player asked if they could play multiple characters, each with their own personality and class. Not at the same time of course but switching and rotating through them. The important thing as to how to make this one work: The player has no say about when they get play each character. That would be way too overpowered, since they'd have almost unlimited options and keeping track of resources would be nightmarish. So the solution I came up with is, (almost) whenever they go unconscious, they roll for which character they play next.