This is a bit out of my regular area of expertise, but I'll try and provide some pointers. Under s65(1) of the Local Government Finance Act1988, it is the person "entitled to possession" who is legally responsible for the business rates.
Until you vacated the premises and wrote to the landlord to tell him you regarded him in breach of the lease agreement, it seems pretty clear you were responsible. So if you were paying the landlord and he failed to pass the monies over, that's unfortunately your problem. They will still come after you and that is why the council are saying it is a civil matter between you and the landlord - to get the monies back from him.
After you left, it is a bit muddier. The issue is whether you are entitled to possession. I'd be inclined to write back to them stating you are not, and use this terminology, and see what happens. It strikes me this landlord is not going to accept to the council that the lease is terminated, even if he has no intention of pursuing the matter with you.