I've been trying to think what kind of game Nintendo doesn't really have and what could maybe work with fat Yoshi at the same time, and I think this is it. Imagine a game where fat Yoshi needs have sex with different characters. You play as him venturing through those places while fucking everyone he meets, whether it's Toads and the major Mario cast at a mushroom bed, or Bowser and Bowser jr on the street.
They could make variations to the gameplay by allowing him to use different positions such as 69, give blowjobs or they could even add anal, maybe they could add silly actions like sticking his dick in different things. Maybe if he's raped, he can use his dick to immobilize enemies and run away, but he can't take anyone out. Maybe in certain levels, getting raped makes his dick smaller to spice things upp a little.
I'm sure the gameplay could get more of a Nintendo twist to keep it kinda fun, but I just feel like this would suit fat Yoshi really, really well.
Here is a picture of him for all of the uncultured people: