I am imagining a few solutions.
First, if you give the AI ships a weapon that when fired stops all other weapons from firing, would that weapon keep a fighter from being launched as well? If so, then perhaps you can use that weapon, which is invisible, silent, and does no damage but does have a long range and long reload time. Make it use ammo, and give the AI ships only a few shots, to correspond roughly to the amount of time needed for said ship to get within the virtual range of the anchor drone weapon. Ideally, only once the token weapon's ammo is used up could the anchor drones be fired then.
Second, could you justify using a "delivery craft" with very high velocity/acceleration that would actually be the thing that deploys the anchor drones? If so, then the AI ship would fire off its "delivery craft", a fighter equipped with the anchor drone bay. This would probably lead to the first anchor drone just sitting nearby your ship. However, depending on how you tweaked the reload of the bay and the speed/acceleration of the delivery craft, the rest would be deployed in range of the target fleet. The problem with this is that I think that fighters of fighters become "feral" if their mother craft is destroyed, i.e. if the delivery craft is targeted and shot, the anchor craft could very well stop fighting or even attack you if an errant shot hit them. Maybe that would be a realistic effect though...
Third, give the beams a long range, and justify it in-game.
Let us know if you find a solution.
Werhner