There's always an alternative: Both clubs have separate stands with separate flags and whatnot, but both stands are next to each other. This increases exposure both for the car and for both clubs involved, and also leads to a bit of healthy competition on who can have the nicest layout etc. Obviously that's simply not going to work at all if the people in charge of the clubs can't even be civil to each other, but in time it can work. All it takes is someone to make the first move, and I'm not pointing fingers or saying that it must be someone from here, I'm just stating an opinion.
FWIW I can see exactly where this all comes from, and having helped run a car club before and had 'issues' with rival clubs I can appreciate that the sentiments can run deep, but the one thing that has to be remembered above all else is that the clubs should always work for the benefit of its members. IMHO, anyway.