Enumerating Planar Graphs of Bounded Treewidth
There is a nice software which generates small planar graphs with respect to isomorphism which might help. As I see one of the problem was to generate non-isomorphic planar graphs and most of those planar graphs (on less than 15 vertices) are of small treewidth.

For checking whether their treewidth is smaller than given value $k$, one way is to use heuristic algorithms to speed up this calculation, just in the case that exact algorithms are not practical. e.g in a planar graph $G$ first we can find a diameter of $G$ and the corresponding path $P$ of length $d$ (which is a diameter). Then find a vertex $v\in P$ which has shortest longest distance ($l$) to any other vertex $u\in G\setminus P$ among all vertices $w\in P$. The treewidth of $G$ is at most $d+l$, if this is smaller than $k$ then we are done otherwise either apply some other heuristic algorithms or run the exact algorithm. 

For less than 3-connected graphs it's also possible to apply heuristics by finding cut vertices and then fixing those vertices and finding the tree width of a remaining graph. But as the number of nodes is small ($15$) if the graph is $4$-connected then the diameter is not big and I think the first heuristic should work there. (I don't know if there is any 5 connected planar graph on at most 15 vertices, but as we know there is no $t$-connected planar graph for $t&gt;5$)

As the size of largest obstruction for tree width $k$  is not known we cannot simply guess the upperbound value of treewidth of a given graph $G$. But it seems that at least for planar graphs it should not be so big (one should give a proof for this).