Counting Hamiltonian circuits in a 3-regular Hamiltonian graph is #P-complete, as follows.
Proof sketch. The membership in #P is trivial, so we will only show the #P-hardness.
Section 3 of Liśkiewicz, Ogihara and Toda [LOT03] shows that counting Hamiltonian circuits in a 3-regular (and in fact planar at the same time) graph is #P-complete. Moreover, their reduction from #3SAT maps satisfiable 3CNF formula to Hamiltonian graphs. Therefore, you can reduce #3SAT to counting Hamiltonian circuits in a 3-regular Hamiltonian graph by first adding one trivial solution to a given 3CNF formula and then reducing it to counting Hamiltonian circuits by using the reduction in [LOT03]. QED.
[LOT03] Maciej Liśkiewicz, Mitsunori Ogihara and Seinosuke Toda. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science, 304(1–3):129–156, July 2003. http://dx.doi.org/10.1016/S0304-3975(03)00080-X