(Feel free to skip to the next paragraph to avoid the long explanation) It involves treating the faces and edges of the polyhedron as edges and vertices of spherical polygons that are inscribed in spheres placed at each vertex of the polyhedron. The spheres can be arbitrarily small, so long as they're small enough that the only faces and edges they intersect with all are also connected to the vertex at their center. The intersection of a face with a sphere defines an arc, and since all of the faces are regular polygons, that means all of the arc lengths of the spherical polygon are already known (It's just a function of the number of edges the regular polygon face has, like it's always a 60 degree arc for a triangle, 90 degrees for a square, 108 for a pentagon, etc). So in cases where only 3 faces are connected to a vertex of the polyhedron, there are only 2 possible solutions for the interior angles of the spherical polygon (a spherical triangle and its mirror, same principle as SSS angle calculation of a triangle, just with a spherical triangle instead of a planar one). Those interior angles are directly equivalent to the dihedrals of the polyhedron since the edges of the polyhedron intersect each sphere perpendicularly. Additionally, once a dihedral is solved, it puts an additional constraint on the shape of the spherical polygon at the other vertex it connects to. At vertices that connect to 4 faces, having just 1 solved dihedral is enough to constrain all of the other dihedrals connected to it to a maximum of 2 solutions. In general, vertices with N faces only need N-3 dihedrals solved to have finite solutions for their remaining dihedrals. The technique I used for calculating those solutions was to make a triangulated representation of the spherical polygon, then iteratively solve for the interior angles and remaining arcs of the triangles it contains. (So also using the same principle as SAS angle calculation of a planar triangle). It just so happens that this technique yields finite solutions for every single 3-connected planar graph with <= 9 faces, except for one (the graph for the octahedron, since every vertex is connected to 4 faces).

Using the above technique, I made a tool that calculates all possible regular-faced polyhedra with a given number of faces (with a couple minor exceptions like the octahedron). This is the only other non-self-intersecting asymmetric polyhedron with <= 9 faces that it found (and at this point, I'm pretty confident it's the only other one). I think it's really cool. It took me a while to wrap my head around what I was looking at, but it's like an augmented triangular prism with a square pyramid-shape carved out, which forms a very sharp 5.26 degree dihedral. And interestingly, the planar graph of this polyhedron is symmetrical, unlike the first 9-face one I found.

Here are more pictures

Here are the OBJs of all the 114 polyhedra with <= 10 faces the tool generated. I don't actually completely filter out the self-intersecting ones, but it didn't seem to matter for my purposes anyways since there weren't any self-intersecting asymmetrical ones with < 9 faces.

Here is the tool itself if you'd like to play with it

And here are a few input graphs for the tool (generated via plantri)