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 Brègman'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 $|V(G)|$). **See the edit below.**

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 $K_{2,2}$.

**Edit:** If the sum of the degrees is fixed, then the product in the upper bound here is maximized when all of the degrees are as similar as possible. This can be derived, e.g., from Karamata's Inequality. In a planar graph, the average degree is at most $6$. So, the number of perfect matchings is bounded above by
$$(6!)^{n/12} = (1.73026\dots)^n.$$

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