A while back, when I went to use my MS branch punch (the one we all call pine bough) I found that I couldn't punch my SU paper any more. Talk about bummed!! It worked on printer paper, but that just wouldn't do.
Well, I was upset for a long time. I tried silicone lubrication, foil, wax paper...nothing worked. Why I eventually tried what turned out to be THE solution, I don't know and I don't much care...but it WORKS.
I took a picture of how I punch with that punch now. Because I had to use my right hand to operate my camera, you'll just have to imagine it being on the punch lever...
Try this and see if it works for you. I have absolutely no problems any more. I put gentle, but firm pressure on the punch. You don't have to lean into it.