I'm inclined to take the MIL's actions a little less seriously than the others. If taking BabyBartfast to Baskin Robbins is a usual thing, then it seems to me not unnatural that while they're waiting for their ice cream to be made one day near BBF's birthday, the MIL might say, "Oh, look, there are some cakes here! You're going to have a cake for your birthday in a few weeks, BBF! Won't that be fun?" BBF wanders over to look at cakes and has fun picking out her favorite, etc. The next week, they're there, and BBF remembers looking at cakes and her birthday coming up and wanders over to look at cakes again. Etc., for each of the next few weeks. It's not so much the MIL pushing, as the MIL seeing the cakes, being reminded that BBF's birthday is coming up, and thinking it fun to let BBF look at them. It may not have occurred to her (depending on whether previous cakes were similar to this one) that the cake would be significantly different... after all, a cake is a cake, right?
I do have to say that, as cool as the cake in the picture here is, for an *older* kid, it is nothing like what I'd expect from a cake for a little kid. I can really see why BBF was surprised and disappointed to see it. It looked black, or close to. And while she might like all those things separately (dinosaurs because they're big and ferocious and make kids feel powerful, ballerinas because they're beautiful and graceful, space because it's exciting), put all those things together and it's a bit... eclectic. I'm not saying the cake is bad, because I can totally see why people think it's awesome, but I think it's coolness is in a way much easier for adults to appreciate than kids. I can see why BBF wasn't really sure what she wanted on a cake, and yet was disappointed with what she got... she was probably imagining the traditional cake with light-colored frosting and little puffs at the edges, and wasn't sure whether she wanted a picture of a ballerina or a dinosaur or whatever on it.
And it does sound like your DH had a valid misunderstanding with what the problem was. He did clamp down on the parts that he probably felt made it like a party, and limited it to only those things that seemed reasonable for "recognizing somebody's birthday, on the day, while out and about." Is having a small birthday cake and having the staff sing "Happy birthday!" so different from, say, when you go out for lunch on your birthday and mention to the waiter that it's your birthday, and they bring you out a dessert and clap and sing "Happy birthday?" So I can see him thinking that it was within bounds.
I also have a similar issue with my mother, who desperately wants to spoil my girls. And the degree to which we let her varies a bit, depending on how far away we live (too many snacks are not as much a big deal during a one-week visit as they are if we're living in the area and she's seeing them every week or so). She's very much a gifts=love type of person, so she always wants to buy them treats, both of the food and of the toy variety. If only I could get her to think that homeschool books=love. Alas. So there's a constant pressure to push back against what we see as "too much" and it can sometimes be rough on the relationship. But if I were in your shoes? I'd probably a) have the second cake be "today is your *actual* day," and not play it as a replacement cake, and be okay with it, and b) if not a, then I'd at least be reasonably satisfied with DH's explanation and agreement to discuss MIL ideas in the future so that you both understand exactly what is okay and what isn't of MIL's ideas.