Typically, a "thin western pad" is a navajo blanket, which actually COULD be folded in such a way to accommodate the flap length of that saddle. But yes, that pad is doing more damage than good. You'd be better off using a fluffy bath towel folded properly than that thick pad, if that's the only option you have. It may have been the pad you were given with the saddle, but it most definitely isn't appropriate to use with it. I would look for other options if I were you. Pads do not have to be expensive, but they do need to fit the saddle.
"Keep a leg on each side and your mind in the middle"