I have to agree with everyone who says it's okay to use many different bits. Quite honestly, I see no problem with a horse being acclimated to work in different bits. As long as you aren't rotating extremely harsh bits (which the OP is not) then it's no big deal. My trainer has me switch bits every 2 weeks during my lessons so that I can get used to working with different bits. We are mostly using snaffles, but I have used a pelham, too. I find that it helps me to remain quiet and calm with my hands as well as learn how to use different bits and what bit works best for my horse.
My horse and I aren't using "stronger bits", actually they are all about the same level of "harshness", which isn't very harsh. It keeps my horse paying attention to me because I've learned that, with him, he needs a change of something every now and then to keep him paying attention.
In any case, the OP is using different bits for different ways of riding. It's like how if you ran barrels and did dressage with your horse you'd use 2 different bits.