Algorithm: Finding shortest path through a dungeon in a game
Travelling Salesman Problem, even if you allow repeating nodes is NP-hard. See Computational Complexity of TSP.

Umans and Lenhart show hardness results for Hamiltonian Graphs in Solid Grid Graphs, 1997.

TSP for Euclideal Case (or graphs with triangle inequality) also imply NP-hardness of TSP with node repetition. TSP even for manhattan distance $L_1$ (or $L_\infty$) metric is NP-complete. See the original Papadimitriou's paper on the topic.

You may be able to prove NP-hardness of TSP for your case by adding arcs to nodes that have corresponding distance as length of shortest path between nodes which will simulate repetitions of the nodes. TSP for your special case looks like an NP-complete problem.

So either write a sufficiently good (heuristic wise) exponential branch and bound algorithm to compute a shortest tour (which may not be all that inefficient, if your graph is small) , or forget about optimization and calculate a good enough approximation.