Dog owners pay council tax and as far as I know Llandaff fields are owned by the council.
The fees you pay allow you to book the pitch for an hour and half on a saturday/sunday morning. It doesn’t give you the right to govern it’s use outside that time slot as far as I’m aware.
If I had a dog I’d use it regardless of what football people or the council say. I’d clean up after me and if you asked nicely I’d take some disinfectant spray and a rag to clean up properly and leave your sports facilities inmaculate.
Alternatively, you could label me and my friend as a nuisance and ban me.
..if that seems more reasonable.