Yes, as far as I know, the companion's quest can only be continued when controlled by whoever recruited that companion. We have the same setup as you guys (3 co-op with 1 companion) and stumbled upon the same issue. There is no solution as far as I know, though I know of a work around, if you have a single person you'd like to control that companion for the rest of the game.
When you complete act 1, whoever you recruited will "leave" your party and you can re-recruit whoever you want. This is the chance to change who recruited the companion, so you can proceed with the quest controlled by another player. The companion is still "locked", just to another player.
No idea why this is the case. Seems like a coding issue that they can't just solve now, since it makes no sense as a conscience restriction to put on your players.