I am looking for references for the following problem: given integers $n$ and $k$, enumerate all non-isomorphic planar graphs on $n$ vertices and treewidth $\leq k$. I'm interested both in theoretical and practical results, but mostly practical algorithms that are possible to code and run for as large as possible values of $n$ and $k$ (think $k \leq 5$ and $n \leq 15$). If you already have an answer, ignore the ramblings below.
The following approach works sort of ok for enumerating all non-isomorphic graphs on $n$ vertices and treewidth $\leq k$ (i.e when the planarity constraint is dropped):
(a) Enumerate all non-isomorphic graphs on $n-1$ vertices and treewidth $\leq k$.
(b) For each vertex $G$ on $n-1$ vertices and treewidth $\leq k$, every clique $C$ on $\leq k$ vertices in $G$ and every subset $S$ of edges in $C$, make $G'$ from $G - S$ by adding a new vertex $v$ adjacent to $C$. Add $G'$ to the list ${\cal L}$ of grahs on $n$ vertices and treewidth $\leq k$.
(c) Trim ${\cal L}$ by removing copies of the same graph.
A tempting way to extend this to enumerate planar graphs of treewidth $\leq k$ is to simply filter out the non-planar graphs at every iteration. Unfortunately this fails to generate all planar graphs of treewidth $\leq k$ (for example because it only enumerates $4$-degenerate graphs).
Of course we could enumerate all graphs on $n$ vertices and treewidth $\leq k$ and only then filter out the non-planar ones, but this fails to exploit that most graphs are non-planar and seems very sub-optimal.