It is quite a technical question that is being asked, and we need to be careful about the distinctions here.
That is, we need to bear in mind that the definition of Public Funds in para 6 of the Immigration Rules includes Council Housing. But it does not include Housing Association housing, unless the accommodation is allocated because of a nomination from a Council.
However if someone applies direct to the Housing Association, and gets allocated accommodation because of a Housing Association decision, that is not Public Funds.
But starcrossed you may have a further hurdle to pass. That is, each Housing Association has an allocation policy and it sounds like this particular Housing Association is trying to exclude those subject to Immigration Control. If so, such exclusion would not be because of the Public Funds issue, but simply because that is what they have decided to do. There are no doubt other exclusions as well.
But you are British? Yes? So absolutely no problem you applying for the accommodation in your sole name, if need be.