Short answer - appeal it or pay up.
Whilst the council might not be enforcing it, it usually is just merely a case of a slower legal process and whether they can be bothered to pursue it, theoretically yes they can take you to a court and have a judgement ruled against you which just makes matters worse and affects your credit, then they can take it further to the high court if you still don't pay. It's more effort than it's worth to a lot of private companies but risk it at your peril.
Appeal first, it'll buy you some more time. If you lose and still can't afford it ask for more time to pay.