Merging two Augmented Red Black Trees (Order Statistics Trees) in optimal time

Question

I've been having trouble figuring out an optimal algorithm to match the lower bound on the following problem: merging 2 red-black trees, sizes m and n respectively, n > m. The red black trees here are assumed to be granted an augmentation, having the subtree sizes as stored information.

Using the decision tree approach, I can obtain a lower bound of m*log(n/m + 1), yet coming up with an elegant algorithm has surfaced many issues.

I have tried the following procedure, but came up with issues when analyzing its time complexity:

First, using the augmentation, select in the tree n the n/m, 2n/m, ..., (m-1)n/m values and store them in an array.

Then, split tree n using those values to obtain n/m different trees.

Then, merge each of the selected values to the smaller tree m inside a new list.

Using that new list, decide in which split subtree of size n/m to insert each element of tree m.

Finally, merge all n/m subtrees back in the original tree.

The main idea is that the middle process of inserting each element in a subtree yields precisely time O(m*log(n/m + 1)), and the merging of the selected elements to the smaller tree takes time O(m)

The big issues appear in the other 3 steps. Using the standard basic implementations in red black trees of select, split, join, it seems that I can't reduce the time complexity of those to anything smaller than O(n).

My question is whether there would be a property of RB trees I am missing in order to obtain the proper bound on the algorithm.