No, you should not have to pay to rent the horse you are considering buying or leasing.
That’s the seller’s investment. And that paid off for the seller in that you decided to lease one, right?
If you paid your pro for helping your kid during those trial rides, and those were billed as “lessons,” there should be no commission paid on the lease. If you paid him a commission for setting up the lease, his helping your kind during those test-rides were his investment in the deal. It’s part of what he did in order to help you guys decide to lease the horse.
Take or leave this bit: I don’t like the willy-nilly extortion of clients that happens in many corners of the horse biz. If I were doing business here (and I’m one of those old ammies who has been around the block a few times), I’d be going to my pro and explaining why I wasn’t paying those $135 fees for the use of the horse I was trying out. If they couldn’t make those charges go away, I’d cancel my lease and walk. They probably don’t want to lose a year’s lease fee, your board check, your training and showing business for a measely $540. But if they do want to do that, your much larger business is there’s to use. I would not compromise on this point both because of the math and because it’s early in the relationship and you want to make it clear that you and the pros will treat each other like professional adults.