Maybe use gloves that are easy on the finger tips or very flexible, like a strong latex, or a couple of layers of regular latex gloves. Maybe use rubber finger covers meant for string instruments, as they are rubber and will help you grip the balloon and wont pop it. Or maybe try deflating them a bit because as you twist, you're forcing more air into different parts of the balloon which causes too much pressure. It will look the same if you just deflate it a tad.
Hope this helps!