Back with another update.
First, for everyone that feels Corsair doesn't stand by their C1 products, I felt the same way originally. Honestly though, I have gotten significantly better support when I talked with someone on the phone than via a support ticket. Link them to my post above and ask them to do the same for you.
Onto the installation. The original AIO was fairly easy to remove. Two screws were a bit awkward to reach, but I had the CPU in a matter of minutes. Two screws to remove the side of the case, 4 screws to detach the pump from the CPU, and there are two wires you need to unplug from the board, don't forget the placement of the small wire. The new AIO came with thermal paste already applied, but that had to go. The paste must have dried out or something because I was getting 100C temperatures when I used it so I removed the paste from the CPU and pump and applied Noctua NT-H2 instead. Now I'm getting CPU temps in the 40-50C range with a max of 70C. My C1 is completely fixed.
I would highly encourage removing your AIO and applying a new set of thermal paste, but also link support to my post and ask to be helped the same way I was.