Pendantry aside ;), would you describe it as the Seat of the Mayors, in which case Mayors’ Seat is correct, or Seat of the Mayor, in which case Mayor’s Seat is correct.
Another thought is that there are not multiple Mayors at any one time (unless it is intended to apply to mayors from other towns) so the latter example above may be more accurate.
Either way, I wouldn’t be taking the decision in isolation no matter how much I thought I was right. Get the buy-in from whoever placed the order for whatever you do. If you want to change it from the requested “Mayors'” option without authorisation, you may face issues, even if you are correct!