I don't know if this will work on your grillcloth, but shaving cream works on car seats when stained and dirty.
Just rub it in and let it soak in a while, then wipe it out/off with a damp cloth.
And maybe the dog isn't a Mesa fan but just a Marshall fan staking claim to what he considers his.
If so, let him believe the cab belongs to him, don't go remarking it yourself and just borrow it when you want to play. That should work.