In Two-factorizations of complete graphs it is stated that $K_9$ has 122 non-isomorphic Hamiltonian decompositions, and the corresponding number for $K_{11}$ is 3140. I don't think they know any other values. (Sloane's database does not have any sequences with these numbers in.)

Now you are interested in the labeled case, which may be easier. However I have not been able to find anything (on Google).

In Two-factorizations of complete graphs it is stated that $K_9$ has 122 non-isomorphic Hamiltonian decompositions, and the corresponding number for $K_{11}$ is 3140 (EDIT: the actual figure is much more than this - see comment). I don't think they know any other values. (Sloane's database does not have any sequences with these numbers in.)

Now you are interested in the labeled case, which may be easier. However I have not been able to find anything (on Google).