Sweet pics!
If you look at the previous page in this thread, I posted a pic of my clamp. I have the same Irwin as you. Different straighteners, but they're all pretty similar. I started by putting the straighteners right in the clamp just to try it out, but then I stripped them down. It's a lot better now.
I took apart my straighteners and attached them directly to each of the clamps with double sided tape. Each heating element already had a solid hard plastic base behind it, so I only had to remove all the extra plastic around those and the handles, then tape that right to the clamp.
It took a small screwdriver to pull a couple of the screws out. Then I used a flat-head screwdriver to pry some plastic parts apart. And finally, cut some parts off with a Dremel tool. It's not too difficult if you take your time and look at it carefully. Be cautious to not break any of the wiring connections while taking it apart. I taped the handle that has the power control and power lead right to the clamp as well. Also, I covered the metal rail of the clamp with electric tape just in case any of the wires ever touch it. The wires that go from the power lead to the heating elements were already covered, but I taped the rail anyway to be certain.
Someone else, maybe
@heady blunts ? posted some pics a few pages back of similar breakdowns that took apart the straighteners and put the heaters right on the clamp as I did. Looking at those pics also could help.
I suggest it. It makes pressing with the clamp much easier because you're just dealing with one 'tool' instead of juggling the clamp plus the straighteners.