Is it permissible for a man to fall short in fulfilling the rights of one of his wives in compliance with his mother, and then give his wife the choice between remaining with this shortcoming or separation, and the wife subsequently stays in this situation?

There is no blame on him if he gave his wife a choice and she chose to stay, and there is no sin on him. The sin and blame are on his mother who forced him into this situation. If he can advise his mother himself or through someone she accepts, informing her that her action is impermissible and that she risks worldly and otherworldly punishment, then this is what is required. Otherwise, Allah does not burden a soul beyond its capacity.