Doesn't really sound like Star Trek
Has all the story elements, just one thing missing, No Federation. The crew are not Star Fleet officers, they do not have Star Fleet training or discipline all they have is a Starship and a captain that's analogous to Captain Bligh. The people on this starship have to work this out together, they're one ship against the galaxy, probably the Klingons won't be much of a threat unless they wish to take on the entire Klingon fleet!
In reality, Captain Bligh was known to be an overly agreeable and accomodating captain to his crew and that his permissiveness allowed the famous mutiny to form and allowed conditions on board to deterioriate to the point they did.