You can answer "what is the maximum edge weight on the path connecting u and v?" by splitting the path into two parts: u up to lca(u, v), and lca(u, v) down to v. For each node, precompute the maximum edge weight on the path to its ancestor steps above.
This extends binary lifting to carry edge weights. During LCA computation, you simultaneously compute the maximum edge weight encountered while climbing. This adds no extra queries.