I am taking out a loan, and the bank requires a mortgage to be registered on my property. For cost reasons I would like the bank to handle the formalities. My notary says this is impossible. Is that true?

Yes. For a mortgage to be valid it must be executed by notarial deed (Civil Code, Article 2409). The notary handles registration and publication with the land registry service. The mortgage then becomes enforceable against third parties.
The notary must practice in France if the property is located in France (Civil Code, Article 710-1).