I ended up implementing a bidirectional, greedy version of breadth-first search, which suits the purpose well enough. To put it simply, I just had the program look through each node its starting node connected through, then each node those nodes connected to, then each node those connected to... until the destination node was found.
Normally one would build a list of appropriate paths and pick the shortest one, but I tried a different method; I had the program run two searches in parallel, one from the starting point, and one from the end point. When the 'from' search found the last node of the 'to' search, the path was considered found.
It then optimizes the path by checking if each node on the path connects to a node further up in the path, and deleting each node in between them.
Whether or not this funky algorithm is actually any better than a straight BFS remains to be seen.