Time to implement the full tour variant. During DFS, record each node every time you visit it (including when backtracking from children). Build a depth array parallel to the tour. For LCA queries, find the first occurrences of both nodes in the tour, then run RMQ on the depth array between those positions.
The node with minimum depth in that range is the LCA. Use a segment tree or sparse table for the RMQ. Test carefully: this variant is more complex than the entry-only version.