Asssuming that you are a UK citizen, in theory you don't need any paperwork to take your boat to France & use it. BUT, we are dealing with the French on their own turf (or sea)
& they can be very errr French. I've been taking my boats across for around 20 years, & in that time I've been stopped at sea by, in no particular order, Douane (customs), Affaire Maritime (general sea police, similar to fisheries protection), Gendarme Maritime (sea police) & the French Navy. I've had the Customs stop me at sea & then be waiting for us back at the tent, which they searched. I've been stopped & searched at Cherbourg getting on to the ferry on my way home. The problem is, that smallish boats like ours are only allowed 5 miles offshore under French Law, UNLESS, you are British Registered. This is actually bo110cks, as an unregistered boat could quite easily cross the channel to one of the French channel ports, & then not be allowed to sail back home under French rules. I've found that having an SSR number & flying the red ensign cuts through a lot of the hassle. Customs still bother you now & then, but I don't actually mind that, at least they are doing their job. This is what I take in the form of paperwork:-
Ships registration doc (SSR)
Ships radio licence
ICC (international certificate of competance) technically not needed, but sometimes they don't know thier own rules & the rules are applied differently in different regions.
Purchase receipt copy (proves boat is mine & that the VAT was paid)
I have actually been stopped at Portsmouth, in the queue for the ferry & asked to prove that the boat was mine by a couple of bobbys patroling the port. They said they were having a clampdown on stolen boats being shipped out of the country,
it's only happened once in 20 years though