Your dog(s) will have 3rd party insurance under your house cover policy.
We had MoreThan on our dog in the UK. When we called to enquire about it's validity (OH mentioned the 'move' word) they cancelled the policy.
I think it is like car insurance - only valid for small time periods.
There are pet policies but I don't think many folks have them. Vet fees tend to be reasonable. Visit is (like people) about 20 euros.
Note I think you may want to get him an identity card : https://www.siev.fr/index.php
I contacted them about my dog (the second one was born here and got one when we bought him) Just ask your vet (they have to fill in the form.) It was 11 €. There was something about 'importing' which should have been done, but the vet did something to the passport and it was ok in the end.
Oh and he will need YEARLY rabies jabs compared to the UK 3 year interval.
Best thing to do Adrian is doublecheck the current policy your dog has. If it's not absolutely clear in there check with the insurers. If you won't be covered then you could ask them what they advise.
Don't know how you'd be fixed for a french insurance policy - a lot dépends on the age of the dog. I don't know about other insurers but I do know that AXA have a very low age limit on dog and cat policies - about 4 (human years) I think!