Atarlost Diagonal facings do not seem to be considered for ships without a diagonal movement facing.
I have turned on 8-way firing arc checking for the next update, because it was easy to do and I don't see why not!
Atarlost weapons that fire into an arc but don't quite reach the centerline are treated as not firing into an arc at all even though they work just fine against targets of comparable size
In the next update, weapons will now check to see if any portion of the target's bounding circle is within their firing arc, not just the target's center point.
One caveat, though, is that they still do a raycast against the target's center to make sure it isn't blocked. (There's not really a CPU-cheap way to do that check against an entire circle.)
With these two changes, your Eagle ship now chooses the correct rotation. (The 8-way check was insufficient due to the placement of those missile launchers.)
Atarlost Shields are not considered. It is almost always better to expose a facing that has lost weapons over a facing that has lost shield generators.
I welcome ideas on how to factor shields into the equation. It just can't be as simple as giving shields an AIFirepowerRating equivalent and adding that to the weapons, because then it might choose a side that has only shields and no weapons.