Modifying Floyd–Warshall Algorithm for Vertex Weights
"I was wondering if the Floyd–Warshall can be modified to handle this, and in particular if the following idea will work."

Yes, your cool idea works! In fact, other people have found the same cool technique of including vertex weight in the new edge weight. By the way, I would not say you have modified the Floyd-Warshall algorithm. It is better to say that you have adapted the problem brilliantly so that Floyd-Warshall algorithm can be applied flawlessly.

"If anyone could envisage a scenario where this might not work or if anyone knew of a better way, I would be thankful"

It looks like there is little chance for you to become thankful by that cause.