| Who knows what will happen?
This answer is opinion-based, due to lack of canonical information to provide a definite answer.
In an online comic at Nick.com, Bumi and Toph do fight, while in the Order of the White Lotus camp. However, they stopped before a winner could be determined, so as to not attract the attention of Fire Nation military, stationed at Ba Sing Se. In the battle, both are very different but both very capable of bending huge amounts of earth. Both also hold the neutral jing tactic, waiting for each other to initiate the battle, rather than carelessly attacking.