They should only be forced to do what they signed on when working for the player then. If the contract says wear XYZ, then they should negotiate it out, or wear XYZ. Since I’m not hearing that displaying XYZ is part of any contract with a player, they should tell the PGA to pound sand, and wear whatever pleases them.
The player pays the caddie. The PGA has requirements for people on the course during the tournament, including the caddie. They require the bib when on the course.
So, they don’t pay the caddie but require him to provide an advertising service.