Thanks for the positive response everyone - Just wanted to post again and comment on a few points:
Not fully no, we tried things in that direction (not using a true PID method but a quicker & dirtier way of avoiding overshoot) in a set of very unscientific and subjective tests it "felt weird". The method we ended up going for is one I was more confident would work in the ways we wanted while being controllable (and more understandable) by design. If I were building one of these things in real life I'd definitely have gone with a similar approach to what you suggest, and if there was time available to try again I'm sure it could also be made to work in the game.
It turned out to be a knotty problem either way though, sadly hindsight only works after the fact... Pesky reality with its "rules" .
And yes the current method does simulate the weapon not being sure where to aim rather than not being able to keep up. Which is a little questionable, but not the worst piece of handwaving in games.
Most RTS games and similar higher-level things do exactly that. But in this genre but you'd lose a lot of the emergent effects that come with simulating things more accurately: the chance to hit being affects by the targets' size, shape, speed, range, facing (note how much smaller target an eagle is face on to you than from above), cover, etc... you could of course go back and add multipliers for each of those factors, but in the end it's not all that much simpler than just using physics. And you'd still have to work out where to put the impact effects & shots that missed to stop them going through things.
The only current option is via the functions panel to flip them back to fixed-forward weapons (currently no keybind for that but it has been requested before).
For performance reasons you'd want to use the ship's bounding shape of convex hull - generating a uniform random point in a complex mesh is not a cheap operation.
If you went this approach you'd still have to make that target random walk around inside the volume so that it moves smoothly for continuous beam & rapid fire weapons. The nuances are different but should work and be doing a similar job to the harmonic wobble of the weapon direction, but would be more expensive to calculate - and when there can be quite a large number of turrets/gimbals in a scene speed is a concern!
I like to call it the process of adding "Plausible stupidity" . Luckily we do have the advantage of deciding which bits of physics we'd rather just ignore, it's very interesting from my side to talk to people building the real thing as well - I had an fascinating chat with a relative who worked on missile guidance recently.
Right - I'm off to my weekend , Fly Dangerously commanders!
Thank you for the reply and great explanation, it's greatly appreciated.