This sounds like the problem is entirely due to the style of riding. If you are getting pinch flats, they likely occur when you ride up over a square curb or other sharp surface and the tube is punctured by the edge of the rim. Normally, if you can't avoid these events, you could minimize them by inflating your tires more. But since you are already inflated to the max pressure, this won't help (and you are in danger of blowing the tire off the rim).

My advice is to learn how to ride to avoid these pinch flat situations, it isn't that hard but you do have to pay attention. Failing that, you could switch to tubeless tires which are more forgiving in this situation.