Check with your insurers because usually, even if a UK insurer offers a policy with 365 day EU cover, it is a condition of the policy that you are UK resident. If you're not, and you don't tell them, they would class it as failure to disclose a material fact, and would be within their rights to invalidate the policy.
Also, under EU law it isn't permissible for a car to spend more than 6 months out of 12 in a country where it isn't registered. (Q"So why do they offer a 365 day green card if it is illegal?" A "Because it wouldn't be illegal to spend 3 months in France, 3 months in Spain; 3 months in Italy etc...)
It is possible register a car in France if you live on a boat, and these are the documents that you need to produce:
Immatriculation d'un véhicule : avec quels documents prouver son identité et son domicile ? - Service-public.fr
"Personne habitant sur un bateau de plaisance
Attestation établie par la capitainerie du port d'une propriété d'emplacement ou d'une location permanente,
Attestation d'assurance pour le bateau,
Titre de propriété ou contrat de location en cours du bateau".
If in fact you will still be considering yourselves UK resident, you might want to get a quote from NFU who offer (or used to offer, I presume they still do) 365 days EU cover as standard with all their policies, no extra charge.
PS Have you read Terry Darlington's
Narrow Dog to Carcassonne? Brilliant book, it had me in stitches.