Julie: What good luck!
I think you are making a questionable assumption, which is that you need to get your mouth around the mouthpiece. Try just pressing your mouth over the hole. You can apply your lips in different ways against or over the the hole (or the end of the mouthpiece) — this is part of technique, sometimes pursing the lips to thin the air stream, vibrating the lips and whatnot. You can experiment. One of the delights of the melodica is that you can do any damn thing you like. The Yamaha mouthpiece does not lend itself to this as well. Here’s a photo of my favorite Suzuki mouthpiece, with which I do get the knob into my mouth (these have to be ordered from Japan):
Here’s my other favorite, which is just the fitting from the melodica end of the long tube, also adapatable with minimal ingenuity (not very well) for a Yamaha:
And this photo shows why the metal tray Suzuki does not lend itself so easily to playing without a mouthpiece. There is a space around the hole, so that if you cover the whole opening and blow, air escapes.
On the whole, I prefer to play the Yamahas without a mouthpiece.