I have done more testing.
Again, I am assuming that the main weapons of ships will not be a decisive factor against drone swarms. These recommendations are for turrets. Equip your ship with the biggest gun possible to kill the enemy capital ships while your AI companions ward off the drones... Or die trying. If you have to decide between more turrets or more firepower, choose more turrets. You need to fill the sky with fire to get as many drones as possible, as your turrets will only shoot at one target at a time.
Beams: My first test was a combination of theory and limited combat (drones spawned directly against the turrets). A new test involved the trading guild sending a patrol over to deal with the pirates. After the pirates won, they attacked the beam turrets... And utterly trashed them. They just stayed out of range and wittled them down. Maybe use on stations if you expect drones and chaff to be thrown towards you or on ships if you can (and want to) maneuver into the swarm. Favourite is Beam/Cannon for faster reload.
Cannons: Remain viable as ever. Cannon/Cannon with punch-through is the best solution and helps against drones greatly. Still gets overwhelmed, but takes many drones with it. Second best on ships (beams are better) and best on stations.
Missiles: Work, but have long reload times and are a bit unreliable due to being able to be shot down. Added bonus of having enough punch to hurt the mother ship or its escorts if those decide to join the fun and to mostly one-shot drones. In the first case, use missile/pulse. If you use it against drones you (again) take cannon as slave (and hope the turret aims good enough) or just a single beam block to give it lock on capability. Avoid swarm missiles, as said before.
Pulse: Im gonna laugh.... Thats ok, right? It might work in principle, but getting close to enough drones to pull it off will be too difficult.
Added advice: If you design a bomber or a missile turret: Dont stick a cannon on the same thing. The cannons shoot down their own missiles if they intersect, which they will. Beams and pulses are bad slaves against drones. Missile slaves would be good but are unreliable at range and wont hit the core as reliably as their focused brethren.