If I were to build a cart it would be about 1/2 the length of the kayak and a few inches wider than the PVC supports that I would use to support the hull. The PVC supports can be longer than the cart as needed.
The kayak should be at pickup bed height to make loading and unloading easy. Make a place for gear storage like the paddle and PFD. If you leave the lower section open, you can roll a lawn mower underneath.
I took one of those metal shelf racks on caster wheels for the garage, cut the “legs” roughly in half to be the same height as my truck bed and put pool noodles on the top shelf. It makes the perfect kayak cart and I put my fishing rods and other accessories on shelves below. It makes the perfect kayak cart!
PVC will give a little to the contour of the bottom, and slide easily off when you need it to, and it will outlast your kayak. Carpet can wear out, retain moisture, rot and mold. Pool noodles won't last as long as any of the above, can stick to the hull if left for a while.
Definitely PVC. I use 3" SDR 35 PVC. It's thinner wall than common schedule 40 so it has a little more flex. It's also coincidentally about half the price of schedule 40. 2" may work if your kayaks hull is pretty flat.
To fasten the PVC drill a 1/2" hole in the top of the PVC tube above where you want to screw the PVC to the wood rack. That hole allows you to get a screw down in the PVC and screw it to the wood. I had to get an extension so I could use my drill. I used self tapping screws so I didn't have to drill a hole first but I think self drilling screws will work too.
Also, to clarify... the PVC could not sit flat on a plywood platform. The PVC should free span 1/2 to 2/3 the length of the hull... this allows the PVC to contour to the shape of the hull and disburse the support. Otherwise, you will develop a flat spot(s) on the bottom of the hull over time.