My thought is that the presentation of the next cake is the important part of this. Because she's four, Babybartfast is a little young to drag out a grudge against a cake unless she's reminded about it, so if MIL presents the new cake as "I wanted to get you this cake because you didn't like the last one" than that's bad but if it's "I wanted to get you this cake because I love you sooo much!" then all would be well.
As to not liking your answer, she asked your DH and got a "yes" then called you and got a "no", so I don't see calling him back as subversive because as a previous poster said, he's a parent too.
"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 don't see it that way. He didn't say anything like "ask Slartibartfast", he told MIL it was okay. And so I see standing firm just because she called Slartibartfast to be not a hill worth dying on. It's possible she asked her son for permission (and he said yes) and then called Slartibartfast to arrange the schedule rather than "moving up the chain". In general, it's a good idea to maintain boundaries, but this particular fight could spin the wrong way. There's nothing wrong with changing her mind after finding out he's okay with the idea, and such a thing would be acceptable even in front of a child because it shows that they discuss and come to an agreement, whichever side that agreement is on.
"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."
Putting her foot down after he said yes has the effect of shutting him out, because calling a veto for something like this isn't a fight I'd want to take on.