Longest simple walk below a certain weight
Without the "vertex-transitive" restriction, the problem is NP-hard, by reduction from Hamiltonian path.

Without the "vertex-transitive" or "number of neighbors" restrictions, the reduction is easy: set $w$ equal to the number of vertices minus one, and assign weight 1 to all edges.  To accommodate the restriction on the number of neighbors, replace each node with a network of nodes shaped like two trees connected at their roots; each in-edge connects to a leaf of the first tree, and each out-edge connects to a leaf of the second tree, and put weight 0 on all the tree edges.

I don't know if this reduction can be extended to handle the case of vertex-transitive graphs as well.