N/D3 Strauss

In a grand tour, each room has exactly two path segments entering/exit. So in the first figure we can start by filling in with dark lines all rooms that have only two possible paths in and out (e.g., corner rooms). Then, we can take another pass over the 144-room house to fill in with dark dashed lines those path segments which, if not filled would force a dead-end or small closed loop. Again, we take another pass and fill in with light dashed lines the path segments forced by the previously deduced path segments. Eventually, and with some trial-and-error, the full unique path can be found, as shown in the second figure.

As an aside, I found it interesting that in at least one case, some information can be deduced based on the fact that the problem states the solution is unique. As shown in the third figure, if the red line in the bottom middle is part of the grand tour, it can always be exchanged for the three path segments in the fourth figure, and the the two rooms along the bottom red segment can always alternatively be joined to the rooms below.