For a camera like the HPX370, I'm wondering about using a couple of these to make the shoulder mount more comfortable and snug:

Ikan ELE-UP3 Universal Pad 3 Pack ELE-UP3 B&H Photo Video

Any downsides to the idea?

porta-brace makes one too. $50 though

If the PB one is what I found online, it's more intended for mounting on rails. But I did find another possibility -- even more expensive -- that might be the ultimate solution:

Camera Comfort Cushion (http://www.cameracomfortcushion.com/)