I think a naive tree search (start from one tile and try adding others with matching edges) might find a solution pretty fast, because most of the search space gets discarded early.
Good point! I think this heuristic works well if the edges between tiles in completed picture are uniquely identifiable or close to that. However from the images there seem to be only 4 "colors" of edges in the puzzle (8 if you count two orientations for each character), so I think this will prune the search space to "only" perhaps around 100k-500k. Still really good, this makes the search space 50k-250k times smaller.