Is it permissible for the second wife to give her husband a choice between her and his first wife, and is this her right?

A woman is not permitted to make her husband choose between her and his other wife, because it is an unjustified request for divorce, or a request for the divorce of the other wife, both of which are forbidden in Sharia (definition). The husband must fear Allah, treat you well, and be just between you both. As for you, you should discuss the matter with him and request good companionship and justice. If he refuses, you can waive some of your rights, such as housing or maintenance, to avoid divorce. If divorce is the only solution, then put your trust in Allah and have a good opinion of Him, for Allah may enrich each from His bounty.