Sylveon gets a 0.10% from me. Other than the ones we have rated, there are several Pokemon with a higher chance. We haven't rated Victini, Pichu, Darkrai, Deoxys, Meloetta, Dragonite, Plusle & Minun (most of whom haven't a chance but still a higher chance than Sylveon), or trainers like N or Giovanni.
This is just a fun Eevee idea that I don't think is likely whatsover. An Eevee with weak, standard attacks. You can evolve Eevee into any one of its Eeveelutions (you select via a wheel that comes up when you hit B, side-B, or down-B). It then stays in that form until you get KO'd and you respawn as Eevee. All Eeveelutions have similar attacks with different stats, side effects, and flavors. They are essentially flavored clones, so while it'd still be a difficult character to create, it wouldn't be stealing nine roster slots. All forms of Eevee, including Eevee would have Swift as an Up-B recovery. Side-B could be a dash attack, Down-B some kind of energy release, standard B, could be unique for each one? A, Smash, and aerial attacks could be various pounces and swipes. Jolteon could be faster and lighter, with electric damage. Flareon could have high attack power and firey damage. Vaporeon could take less knockback to represent high HP and do watery damage. Etc, etc.
I think a more likely Eevee idea would have just a couple of Eeveelutions (Umbreon, Espeon, Sylveon?) with this idea OR just allow you to summon one or all of them for Final Smash. I think Eevee in general has stayed relevant and popular enough to garner maybe a 5-10% chance. It's probably worth rating in the future.
predict: 5.5%
Victini x5