What is a proper size air conditioner for Texas 384 sq ft?
I would think that 12 kBTU/h would be fine even in north Texas. One interesting feature of window units is that the chilled condensed water from inside is used to cool the bottom coil of the condenser before the condensate is conducted out to the ground. (At least this was the arrangement in the early 1970s when we had a window unit in north Florida.) Central units just dump the 40 deg F condensate to a drain and so waste the cooling capacity of this chilled water.

If there is a separate bedroom, you might consider a mini-split ductless with one condenser unit supplying two air handlers. This would cost a lot more, and require professional installation, but would almost certainly use less power and cool better. They are advertised as employing "inverter technology"; some of these minisplits have a claimed SEER of 20 to 30 and they are available as heat pumps which would heat the cabin in winter. But personally given the price difference I would first try the window unit.

If you put in a window unit, you could cut a hole in a wall rather than use a window and there are even "window type" units (I think called slide-in) which are designed for this type of installation. Some window units can't be installed in a hole in a wall because the wall is so thick that it would cover the intake vents for the exterior part of the unit.