As you have already said, the VC dimension does not depend on the algorithm used: the VC dimension simply characterizes the hypothesis class, i.e. the set of classifiers considered by a learning algorithm. Then, if both algorithms consider the same set of trees, the VC dimension must be the same. In your case, if you impose to CART and to your optimal tree learning algorithm that they learn trees with depth at most h, then the set of trees considered are the same, regardless of the algorithm chosen.
There is a distinction to be made for the VC dimension of the individual returned tree. Indeed, each decision tree class (i.e. the set of functions that can be realized using a fixed tree structure letting the internal nodes and the leaf labeling free) has its own VC dimension. If the two algorithms return different trees, then they likely won't have the same VC dimension, and one can expect that the individual VC dimension of the CART tree will be greater than the other, since it is not optimal. This fact can be leverage in practice, but it is not in CART. For more information on this subject, you might be interested in this paper. (Disclaimer: I am one of the author.)