In order to get an upper bound on $\alpha_4$ (which is probably far from being a tight bound), you can use the Kahn-Lovász Theorem (a generalisation of Brégman's Theorem). The Kahn-Lovász Theorem says that the number of perfect matchings in any graph $G$ is at most
$$\prod_{v\in V(G)}(d(v)!)^{1/2d(v)}.$$
Its not immediately clear to me how to obtain an upper bound of the form $2^{c|V(G)|}$ from this, but it looks like it might be possible since planar graphs have at most $3|V(G)|-6$ edges (and therefore the sum of the vertex degrees is at most linear in $|V(G)|$).

Like I said, this is unlikely to be tight, but at least it might give you a bound. The Kahn-Lovász Theorem is tight for *general* graphs, but I think that the unique tight example is a disjoint union of balanced complete bipartite graphs, which is not planar unless every component is isomorphic to $K_2$ or $K_{2,2}$.