How to stop pinching

You have to move your bits out of the way before sliding metal parts together. And if there are pinching hinges, then you put a sleeve around them (silicone or latex for example).

 

You've reminded me that a small flap of leather was the solution that worked for one of my early cages. It was the older single post and padlock design, so I punched a hole for the post, which kept the leather patch in exactly the right place all the time.

You could do similar, punch holes each side to put the pins through, and have enough folded underneath to stop skin getting anywhere near the pinch points. I'm thinking the leather folded (in a U shape from the side view) so there's a soft cushion against your skin, and the two edges coming up have holes each side to fix it's position on the pins. To save leather, you could experiment with cutouts of paper until you get the perfect size and shape, and then you'll have the paper template to cut your leather patch.