Source: This is all surmise, based on having used Brittany Ferries a lot and seen how they operate
I do not believe Brittany Ferries will care one bit whether you are the registered keeper of the car or not.
Consider this: When you book, you can specify the registration number, but all this does is fills in the dimensions. You can also say "I don't know what vehicle I'm going to bring", and supply the dimensions yourself. It's entirely possible that BF won't know the exact vehicle until you actually check in. So long as it matches the dimensions you said, they won't care.
Consider also this: I have never been asked for any evidence of the registered keeper, or of ownership, or even that I have permission to drive. I like to think that if I turned up in car that had been reported as stolen, they might say something, but beyond that I don't believe they do a live check with DVLA or anything like that.
However
Regardless of what BF want, bear in mind the general rules on driving abroad. Now, it doesn't mention borrowed (rather than hired) cars here, but nonetheless I'd play it safe and take the V5C, as well as proof of insurance and the letter from the owner. Remember also all the equipment you must have in the car. The French police love imposing on-the-spot fines on drivers of UK-registered cars that have committed minor infractions...