Either you, your partner or both of you are entitled. Who gets the money is a decision that either or both of you can make.
If your partner or a flatmate flits owing stacks of Council Tax you, soley you would be responsible for paying it of they couldn't find them, or even if they could find them and they didn't feel like paying. Rudeboy was in exactly that position recently was he not?
You can claim it, but you should claim it honestly, theres no contract between you and your ex and its not the council's place to act as if there is one. Either of you can claim all of the money with out without the knowledge or the approval of the other, just as either of you can pay the bill with or without the knowledge of the other.
Committing fraud to claim small sum of money is nuts, committing fraud the claim something thats actually yours… how nuts would that be.
As I say, forget you said your GF lives there, let them forget it too, write to them and claim the money for yourself. If they ask about your GF tell them she's gone and she's non of their business.