Don't take me the wrong way here, I am not trying to get you to buy one of my browbands - but if you are leasing him and you want to use your own bridle, you can use your own bridle. The only thing you would keep the same, is the bit.
I bet the only reason why he has Nylon tack, is because the BM or BO didn't want to spend the money on leather.
I don't see why it would be a problem to use your own leather bridle.
If you don't mind using the Nylon, which I wouldn't see why you would, then don't worry about it :)