Do you already have a declaration of parentage in the UK? Is this something you needed to do to start the international child support application?
This is actually really interesting to me from a legal perspective because there are so many permutations.
The starting point for you is that he is not on the birth certificate and does not have parental responsibility, but you want him to be recognised as your child's biological father for the purposes of child support and possible other benefits.
It seems like the ideal situation for you would be for him to be formally recognised as your child's father by France, but stay off the birth certificate and not have parental responsibility in the UK. This way you would retain full decision making power in respect of your child until they are a legal adult, but the father would still have an obligation to pay maintenance (whether or not it can be enforced in practice is another matter), and your child would get the full benefits of being recognised as the child of a French citizen, namely French/EU citizenship and the right to inherit.
Now, with regards to establishing paternity where the father contests it, I believe the situation is broadly the same in the UK and France. Basically, you can name someone as your child's father even if he is not on the birth certificate for the purposes of your maintenance claim, and the father can either acknowledge paternity or contest it. If he contests it, the court will most likely order a DNA test. If he refuses to cooperate and do the DNA test, the court may make an adverse inference against him, that is to say they may choose to assume from his refusal to do the DNA test that he is the father and is trying to evade his legal responsibilities. Because if you do not wish to pay child support and you genuinely believe you are not the father, or may not be the father, why wouldn't you do the DNA test which might prove you are not the father and absolve you of any further responsibility?
It would be more advantageous for you if he had to do a court ordered DNA test in France, because that would establish a filial link which would be recognised by the French state, giving your child their full rights as a French citizen, without giving the father parental responsibility in the UK.
However, that isn't taking into account the logistics of getting your child to take a DNA test in France, which may be difficult for you depending on your finances and what sort of disability your child has.
In fact, a better outcome for you would be for him to either acknowledge paternity in France or for the French court to make an adverse inference against him and assume that he is the father based on his conduct. Perhaps the fact that he has filed an application for joint custody might be enough for the French state to consider that he has acknowledged paternity. That would be fantastic news for you because it would mean that his silly game playing has well and truly bitten him on the arse.
If the French state recognises the filial link between them, he will be legally your child's father whether he considers your child to be part of his family or not, and they cannot be disinherited. He can write whatever he likes in his will, but if your child does not get their legal share of his estate when he dies, the French court will have to disregard the will so your child can inherit their fair share. So quite apart from the citizenship issue, it is worth keeping close enough tabs on this man in the long term to make sure you become aware as soon as possible if he dies.
He really has been very silly. If he didn't want to leave your child a share of his properties in his will, he should have kept his mouth shut, paid maintenance without you needing to make a claim, and hoped that the French authorities never became aware of your child's existence.