So you only want dead ends to have loops? You might try computing the depth of each node, marking the solution, and then assigning each branch off of the solution a unique color.

At that point knocking out walls only within the same color won't interfere with the solution.

Alternatively you could take care to track depth and knock out walls between different colors only when the total resulting path length would be greater than the existing solution.

Just go try stuff! All of the examples on Bostock's page that I linked earlier link to JS implementations that you could fork.