I've got a few ideas.
For the time, I think that around 2260 would be good. Just before Kirk's five year mission.
Perhaps, the "Sioux aliens" aren't native to the system at all. What if they were nomads native to the area around the system? For whatever reason, their claim to the system is shaky at best. Because the claim is so tenuous and because of the pressures of the cold war with the Klingons, the Federation just goes ahead and moves to get at the dilithium and whatnot in the system. They are technologically unadvanced (say Earth 22nd century or less) and dirt poor. They are too proud to take the Federations offers of handouts. They know of the system's vast resources, but have lacked the ability to exploit them. When the Feds come along, the "Sioux" want a piece of the action and that sweet, sweet Federation technology and have no problem with the idea of taking it.
Lightly armed or unarmed freighters, mining vessels, and asteroid colonies are no match for even the low tech means of the aliens, so the Federation sends a patrol ship to the area. Unfortunatly for the miners in the system, Starfleet doesn't take the situation all that serious. They send a ship that's pushing fifty years old and not really up to the task of protecting the colonies and ships in the system. The ship is small, not too quick or maneuverable, and doesn't boast the best weapon systems around. It's better than anything the aliens have, but not by much. The Starfleeters have a hard time keeping the aliens out of the colonists' collective hair.