Well, I mean, if you want to give Birdo the copy ability, you might as well say Kirby, then.
But let's be realistic here: Wouldn't Birdo want to be the star of every Nintendo game? It's like Wario, in a sense. You'd have a story driven game, a strange, ambiguous genie offers to grant Birdo a single wish, and that's what Birdo wishes for: To be the star of every game, to have a chance to earn the fame and celebrity of Mario, Kirby, Link, and whatnot.
Instantly, Birdo's life becomes chaotic. Yoshi's been Bowsernapped (Or maybe one of Birdo's eggs?) It heads out to rescue Yoshi/the egg, and the first world is a 4 stage Super Mario Bros. parody of some sort. At the end of each stage, there's funny dialogue, and after defeating the Bowser parody, something happens, and suddenly Birdo has to enter the F-Zero Grand Prix.
That's honestly how a Birdo game could be very successful, IMO.
Edit: I should say this would be Birdo's foothold game. This would be the game Nintendo would use to make it so future Birdo games would be embraced, rather than pushed away. Games after this could be anything, but this game would be used to further define Birdo's character, and have high sales results.