You've discovered one of the pit falls of some of the modern recoil pads, they'll catch on your clothing. These pads are designed to stay put, once the gun is mounted, and reduce felt recoil. The new, softer materials are more apt to snag than the old pads.
Rather than alter your gun, have you considered modifying your mount? I usually shoot targets with a mounted gun, but have shoot low gun Skeet. It's easy to get your gun tangled up in your clothes if you use a straight up mount. I learned to use an out, up and in mount. I got more consistent mounts by having the final gun action into the shoulder, not stopping your upward slide. If you get used to bringing your gun straight up, instead of an outward swing, then adding a layer of clothing may snag your mount. Also, you might consider an International-style shooting vest.
If you're serious about changing your pad, you might first consider putting some tape on the existing one to make it more slippery. Or, another solution is an old school lace-on leather butt cover. Of course, with a smoother butt, it's more likely to slip after you get a good mount.