I do think Eevee would have to come with more of an assist gimmick though, with the Eeveelutions coming in with Special Moves, throws, and what have you. Otherwise, you'd have to make Eeveelution temporary (which I believe has been a thing in one manga and that's it), making the stones act like more generic power sources, or permanent, which is just uninteresting, as it would boil down to which Eevelution does the best in the current matchup.
Maybe you could combine the ideas? Like, maybe Eevee comes with an entourage of Eeveelutions. Using the Down Special would allow you to switch between Vaporeon, Jolteon, and Flareon, each with their own specialties, and their own meter. The meter will run down as they use attacks, and you have to take them out of play to recharge them. If you let the meter run out, this takes longer, and you lose access to them.
Each of the three has their own Eeveelution set (for example, Vaporeon could have Sylveon, Jolteon could have Leafeon and Glaceon, and Flareon could have Umbreon and Espeon) that can be used as assists through their Neutral Special. The two pairs are merely visual, and do the same thing as each other (ex. Umbreon uses Moonlight, and Espeon uses Morning Sun, both heal...though I wouldn't do this specifically), but each set does something different. The Side and Up Specials use the Eevelution's abilities.
Eevee itself is a jack of all trades, but is slightly undertuned to balance out the specialized power of its evolved forms. It's Neutral Special could be something like Trump Card, which grows in power depending on the number of unique Special Moves you have used during the match (or stock if that's too powerful), not including Trump Card itself.
Might could work. Might be weird. I dunno.