Existence of Hamiltonian cycle in a 3-regular $C_n$-free graph
Have a look at this paper "Not Being (Super)Thin or Solid is Hard: A Study of Grid Hamiltonicity" Theorem 6.1 states that HCP (Hamiltonian Cycle Problem) for planar graph of maximum degree $3$ and girth $g$ (i.e. $C_{g-1}$-free) is $NP$-complete, for arbitrary $g&gt;6$ (i.e. $g\geq 6$ in your notation).

So, in your notation, we are left with the undetermined cases of $g=4,5$.

So if you are willing to relax $3$-regular to $\Delta(G)=3$, then $g$ (actually $g+1$ for the girth value) can be arbitrarily large.