IANAL but I don't think that is how trademark works. I believe you're allowed to use any trademark as long as it is being used to describe the thing that it is registered for, you don't infer that you're associated.

For example I'm pretty sure anyone could recreate the Pepsi challenge without asking Pepsi or Coke for permission, as long as it was actually Pepsi and Coke in the cans, and they made it clear that they are not affiliated.

Right, but if I were to make baseballs and use their trademark to brand the balls, or start a league named after the trademark...

Just using their bats with their name visible isn't the problem, of course.