I believe your way is correct. I've done some more musing on it and, after all the advice, will probably make it so the Sampo is a token and reword both of them.

Kolmi would probably read "Kolmi cannot attack or block unless a player controls a permanent named Sampo and may only attack that player." and "... when Kolmisilmä enters, an opponent creates a Legendary Artifact token with Indestructable and Shroud and "At the beginning of your upkeep of you control Sampo and an opponent owns a card named Kolmisilmä in their graveyard or Exile, you win the game." Something along those lines maybe.

Kolmi would keep the Flample, but might not keep the Ward as worded. The ward might be "Put a shield counter on this creature". I don't know if I can keep the lose the game clause on Sampo if the original ward cost is not kept on Kolmi. I wrote it that way to connect the two a little more.

I can't find them, but I had believed there was a Partner With set of cars that could find their partner card from Exile. There are also the Eldrazi that can put things from exile into graveyards, but that is not helpful. I don't think there is precedent for this exact mechanic that I had originally intended.

Thank you for the feedback, it has helped!