Are the minimally 3-connected graphs (edge removal pushes you into a 2-connected graph) the same as the class of 3-connected graphs with the minimal number of spanning trees? There is a nice classification of the minimal 3-connected graphs (or maximal not-3-connected graphs, I forget which direction he does it in) in Jonsson's book "Simplicial Complexes of Graphs".
I think there's also a classification of minimal 3-connected graphs inside a paper by David Fisher, Kathryn Fraughnaugh, and Larry Langley which could help ["3-connected graphs of minimal size".] At the very least, they note that all 3-connected graphs on n vertices have at least 3/2(n-2) edges, and produce graphs achieving this bound. All of your graphs with the minimal number of spanning trees must be one of these minimal 3-connected graphs (as the addition of any if your hypothetical "minimal spanning tree" graph wasn't also a minimal 3-connected graph, removing edges increases to a minimal 3-connected graph would reduce the number of spanning trees.)