The species where warp-capable but the war has destroyed all of that technology a very long time ago.
This sentence, all by itself, basically throws the Prime Directive out the nearest window...
...Also, this star fairing species (one faction) has taken a Starfleet captain prisoner, maybe even hostage, after shooting down a federation shuttlecraft. That's an openly hostile act against the united federation of planets. Possibly even an act of war...
...The Commander is now on much firmer legal grounds for her actions.
I concur. Given the Federation's nature, there'd probably be some sort of investigation. After all, a foreign leader was killed. Perhaps even an official Board of Inquiry. But the Commander's actions certainly would appear to be justified in these circumstances.