You are a French tax resident with no tax obligations in another country"
I assume this is the problem, right? You will only became a tax resident when you live 180 days in the country (check the number of days). This means that in the near future they will ask you to prove that you are tax resident.
If you check the documents to open an account you just need:
- une pièce d'identité en cours de validité (Eng: a valid identity document)
- un justificatif de domicile récent (Eng: proof of recent domicile)
- les derniers justificatifs de revenus pourront vous être demandés. (Eng: You may be asked for the latest income receipts.)
I moved between EU countries and between EUR and non-EUR, and even because of business had to go to another country for opening an account (without even had lived there) and never had troubles.
Anyway as an immigrant/expat it's always better to do this kind of stuff in person, as the online templates are usually for the most common client - the natives.
The next one in the search results was “Credit Agricole”. To open a bank account, you need to enter your postal code
Off course, to where do you want for them to send the letters?
and it only accepts French post codes.
Again, go to the physical location and they will let you open. I have done that (not with those banks and not in france).