Other sports do mandate payments and contracts with companies for their equipment to be used as well, its not just bowling.
The NFL just signed a huge contract, which including a payment to the NFL, with NIKE to be the official apparel company for the teams uniforms and sideline gear. The company prior to them was REEBOK and they were outbid this time around.
This is totally different. You are talking about uniforms. I'm talking about equipment.
Let's ezuate this to Golf, where there are many golf club companies, sponsoring players and providing equipment. Yet, none of them have to pay juice to the PGA. And, golfers are not allowed to have labels, insignias, etc on their shirts. They may wear a golf cap or a tagged shirt, but not the names of their clubs, balls, grips, shoes or any other form of advertising.
My nephew is a Pro Golfer, sopnsored by Callaway. He does wear Callaway labeled shirts and visors, and of course, uses his free set of callaway clubs. but, Callaway doesn't have to pay the PGA for his use of equipment.
If a bowling company wants to sponsor a bowler, they should have that right without any payment to the PBA.