So, I was thinking... what if we did the following:
Just like in the OP, each "crew-member" has a view of the same ship piloted by the Captain.
The Captain is the only one who commands Bridge Officers, i.e. activates BO abilities.
Ships that are populated by players who are "crew-members" cannot join an instance with "normal" ships, so the game offers either the current form of play, or a team composed of one ship with a crew of players.
Each player crew-member is in charge of several mini-games that activate automatically when the Captain uses BO abilities.
Each time a mini-game activates for a player crew-member, the player can dismiss the mini-game or play through it.
Mini-games that are failed, cause the cooldown on the BO ability that belongs to the mini-game to lengthen, while mini-games that the player won cause the cool-down to decrease!
Mini-games that are dismissed by the player, do nothing to the cooldown, thus allowing the player "crew-member" to prioritize which BO ability they would like to shorten the cooldown for.
Because multiplayer ships should have better performance than singleplayer ships, missions difficulty should be Elite, and whenever crew death occurs, there should be a random chance that player crew members will suffer injuries while on duty.