Is it permissible, and not a sin, for a woman to expose parts of her body to a male foreign nutritionist for the purpose of obesity treatment? And will Allah forgive her if she repents for doing so?

A woman is permitted to expose her 'awrah to a male doctor only out of necessity, if she cannot find a female doctor to treat her. As for sending pictures of a woman's body to a male doctor for cosmetic purposes without a genuine necessity, this is impermissible. One must repent from this by feeling remorse, ceasing the act, and resolving not to return to it. Allah accepts repentance and loves those who repent.