On the parenting note, I would want my child to learn how to deal with disappointment, and to learn that there isn't a consolation prize for every disappointment, and their SURE isn't a "do-over."
And if this disappointment is what it takes for her to learn how to be more vocal about her preferences, that would be a good outcome, to me anyway.
(She may have had this image that you'd *surprise* her, that it wasn't as much fun to "order" her birthday cake; my son thinks similarly about presents, thought not about the cake. If that's the case, this is a good time for you to explore the b'day cake problem and figure out a way that she can drop you hints, or that you can "catch" any hints she does drop.)
Etiquette-wise, I wouldn't say or do anything further. With MIL, I think it's much the same as w/ a little kid--preserving your authority is worthwhile. She felt she needed your permission, and she thought you wouldn't give it, so she went to Dad first (OK, he's her son, so maybe it was just that). Then she called to get your permission.
Both she and DH thought that you should be the final choice, right? And you made your choice. Changing your mind now will be just the same as giving in to a kid after they've thrown a tantrum. So don't change your mind.
I'd be sounding out DH about the "coordination of authority" business--did he think he'd given permission? What things does he want to be the final decider on, etc.? I wouldn't want him to think I was shutting him out. So I'd try to explore that.
I think the fact that she got in a snit about it actually tells something about her motivation. She doesn't want to just take your DD for ice cream; she wants to be the "fixer" who does it right, and a part of her knows that her motivation isn't the right one. That's why she asked, and it's why she's upset that she doesn't get to do it.