I just wear gloves. Plenty of light weight breathable ones available for summer. The one time I forgot them my rubber reins gave me blisters in moments (my horse's mouth was fine, however)
Have you been riding with gloves previously? Back in the old days when we wore riding gloves to keep our hands warm rather than for safety, I used to get blisters every Spring when I stopped using them. It was just the reins rubbing on the tender skin between the ring and little fingers (not pulling on the horse's mouth because we never took a contact! ) and after a week or two our hands toughened up and we were fine again.
I second the suggestion to get some lightweight gloves though. Safer and more comfortable than blisters.
I've very light hands - but when I'm actually working a horse I get blisters. Surely by the very mechanics of allowing the reins to slip through your fingers (friction)- ergo blisters, you are categorically NOT ripping the horses mouth! (Just my 2 cents).
I get blisters on my ring fingers from friction on the reins AND I wear gloves. I've found the only solution is to layer up plasters on the ring fingers. My skin is really fragile for some reason.
I would just get used to wearing gloves, you can get some very good thin ones nowadays.
I used to be able to ride without gloves, had a break and my hands softened up a bit. Got back into riding and holding reins gives me blisters (leather reins are ok but I prefer grippy ones!).
I have become so used to wearing gloves now that I feel naked not wearing them.
I admit I can't get to grips with this problem. The only time I have worn gloves when riding (50 years) is when it is cold or I am competing. I don't use the rubber grip reins as they are to bulky for my small hands, I use plaited leather or similar. Perhaps you should ditch the rubber grip reins ?