I dunno… I kind of feel for Tom on this one. It sounds like The Cake is a pretty big deal and that it is a central part of the celebration. Yes, you technically are supposed to go to a party for the company and not the comestibles, but I can see why he might feel a bit unwelcome if this has become a "come over for The Cake" event and he can't participate in that part. Tom was in the wrong for asking her to totally change the menu, but Tina shouldn't have been so grudging about offering an alternative. If she really wasn't willing to, she should have just said, "Sorry, Tom, but I'm going to serve The Cake. I'm afraid I don't know much of anything about vegan baking, but it's fine if you want to bring something."
I guess I see it sort of like inviting a teetotaler to a whiskey or wine-tasting party and then grumbling about making sure that you have soda or juice or some other non-alcoholic beverage. Yes, it's their choice to not consume alcohol. No, you don't have to cut alcohol from the menu. However, I guess it comes across as slightly inhospitable, like, "Hey, wanna watch us do something awesome? No, I wasn't planning on having a related activity that you could partake in, why do you ask?"
To repeat: I do not think that Tina needs to not have the cake, nor do I think that Tom should have asked her to completely cut it from the menu. I simply think it was ungracious of her to make a big deal out of providing an alternative. Much better to leave it at, "I'm sorry, that won't be possible."