As a composites scientist, I would suggest to you to use unidirectional tape prepreg instead of different plies of woven 2 * 2 carbon cloth. Your cloth will be way more expensive than UD prepreg and will result in a product that is unnecessarily heavy and inefficient.
When you use a woven cloth, remember that the fibers in the fill direction will not be contributing towards the strength of the lever when subjected to load during the removal of the tire. So if you use multiple plies, which are expensive, you are using multiple plies with fill fiber that adds weight without contributing to or improving mechanical performance. If I were you, i would only use two woven plies, i.e. the outside plies with a few UD plies in between. With UD, you can make it lighter, stiffer, and cheaper.
When you use a woven cloth, remember that the fibers in the fill direction will not be contributing towards the strength of the lever when subjected to load during the removal of the tire. So if you use multiple plies, which are expensive, you are using multiple plies with fill fiber that adds weight without contributing to or improving mechanical performance. If I were you, i would only use two woven plies, i.e. the outside plies with a few UD plies in between. With UD, you can make it lighter, stiffer, and cheaper.