I agree. You did the right thing telling the Mom.
It's never an easy thing firing a band member. No matter how tactful and diplomatic you try to be in the process, it is something that will almost always be taken personally. More so because of your friendship with the Mom.
What's more important to you? The continued success and continuity of the band? or your friendship with the Mom? Once you've answered these questions, the decision and choice are clear cut. Bite the bullet and move on. The Mom and daughter will get over it, and you all will have learned a thing or two in the process.