Restricted Cast, who would you pick?

I'm going to answer from a pure storyline standpoint. The characters that you really need in your party in order to drive the plot forward are:
Terra - Connection with the Espers
Locke - Connection with the Returners
Celes - Connection with the Empire

That leaves room for one more permanent character. Shadow, Mog, Gogo, Umaro, and for the most part, Gau, are already optional (After the scenario split, you can choose to have Gau Leap and then never recover him). Setzer can totally fulfill his role as an NPC. Strago and Relm provide an interesting twist in the story, but they are not what I would consider essential. That leaves 3, who all have about equal footing in my book:
Edgar - He is the most important to the story, and has the most "screen time". His technological prowess also works well in the semi-steampunk world. His role as the King of (neutral) Figaro also allows him to be connected to both the Returners and the Empire. However, as has been pointed out, he's a King. He doesn't really need to join you.
Sabin - Sabin's role as a wandering monk allows him the storyline flexibility to join up. However, even though Sabin was my favorite character growing up, he really has no importance to the story, other than enhancing Edgar's backstory.
Cyan - He definitely has reason to join your party, as the poisoning of Doma is a pivotal event in the plot. As the retainer to Doma, he represents the people caught in the crossfire of the impending war... someone who was aligned with the Returners, but the Returners didn't get there in time. His people were crushed by the Empire, using the power of the Espers. He has reason to hate all 3 of the essential characters, and yet he also has reason to travel with them.

So, I think I'd actually give the 4th character slot to Cyan. I think it works best with the story, and gives the characters a reason to be involved in portions of the plot that they otherwise wouldn't. Gameplay-wise, he also balances out the party by giving a heavy armor/high damage character.

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On further thought, Terra and Celes are barely ever in your party at the same time in the World of Balance. Storyline changes might be required to accommodate the 4-character structure, such as having Terra "recover" before your initial trip to the Empire and the Magitek Facility.