Why did God agree to banish Hagar?

On the first day of Rosh Hashana, we read about the banishment of Hagar and Ishmael. In the received text Sarah urges Abraham to do something unthinkable, and God co-signs the suggestion. 

As per always, the sages were alert to this problem. Let's examine the brilliant solution. 

In the Midrash (Pirkei de-R. Eliezer 29), Judah b. Tema states: "Sarah said to Abraham, 'Write a bill of divorce (get gerushin) for the handmaid, and send away this handmaid." In the Targum (Pseudo-Jonathan), "He sent her away" is rendered "He dismissed her with a bill (gitta)."

As a man of means in the ANE Abraham could have done basically anything he wanted. He could have kept Hagar around as a concubine. He could have added her to a harem. He could have sold her into slavery. But in this midrash, at least, Sarah is having none of that. She says, "Do this right, and by the book. Make sure she is free and clear. Don't make this sad situation worse. Release her fully, completely."

And God's response to this urging from Sarah? "Listen to your wife"