Hi there! My DD also had a fairly high palate. It did make nursing tricky, especially during the first few months. We had supply problems due to her difficult latch and I ended up with badly cracked nips. But I want to encourage you to stick with it! A lot of the problems we had in the beginning were due to the small size of my newborn's mouth. As she grew, and her mouth grew, she was able to latch on deeper and when that happened she stopped hurting me and she was able to transfer milk more effectively, at which point my supply problems were a thing of the past.
What are you doing about your supply issues? When I had problems, the thing that helped me the most was going to see an IBCLC. She helped me improve my latch, recommended some helpful herbs (fenugreek and blessed thistle), told me to ask my midwife about Reglan (a prescription drug which can increase milk supply- please do not take it without consulting your health care provider!), and rented me a professional scale and a hospital-grade pump. By weighing my baby before and after nursing, I was able to determine exactly how much milk she was taking in, which in the beginning was not enough. Frequent pumping with the hospital-grade pump helped me to increase my supply in a hurry.
HTH, and good luck!