To me, this doesn't sound like flycatchers but an interest in catching and snapping at things even very small, that he actually sees. He has been encouraged in this behaviour now for several months -- when he 'performs', the family responds with some delight and amusement and he gets attention and encouragement to keep doing the behaviour.
Flycatchers refers more to an obsessive behaviour which can be obsessive/compulsive, or may be the result of some neurological trigger. Dogs with it tend to snap at things they cannot see.
But you dog could have a bit of obsessive behaviour with this as well, and because he has been encouraged in it, it is now more fixed.
Two things you can do:
1) stop rewarding the behaviour. So no more laughing at it, encouraging a performance, teasing with noises (he may now see the noise as the 'reward' signalling he has behaved properly), etc. Also, don't scold or talk to him telling him he shouldn't be doing this, etc etc -- even saying 'no' becomes a reward for many dogs as he now has your attention. Totally ignore the behaviour which means don;t even LOOK at him when he starts... Alternatively and also, you can:
2) refocus him on appropriate behaviour and praise. This means as soon as he starts in, produce a favourite toy and throw it for him or distract with a nylabone or similar, and praise in a cheerful voice the minute his attention shifts to the item. If he really does the snapping etc a lot, you may want to carry a couple of toys in your pockets all the time so you can instantly distract and refocus.
Finally, if you haven't done an obedience class with him I would do one as the purpose is not just teaching your dog some cues for behaviour (eg sit, down, heel) but also new confidence and self control and working his mind as well as body. Look for a positive-training class that is based on rewards, not punishment, and does not use choke chains. This will give you a third option when he starts this kind of behaviour -- asking him to go into a down/stay or giving a 'relax' command so he refocuses.
If the behaviour continues and seems really out of order or is happening when there doesn't seem to be anything there at all, you would want to go back discuss this with your vet to be sure it doesn;t indicate an underlying medical problem of some sort.
Also you are right, flycatchers is believed to be more common in CKCS than is many other breeds, so it would be good to keep an eye on the situation overall. I'd consider keeping an event diary so that you can record what triggers the sessions and if you can definitely see some stimulus or are guessing that there might be something there. That should either help you avpid the situations that set him off (eg get him a drying coat and stop blowdrying him; let his coat dry naturally) or alternatively will give you a list of times and triggers for the sessions that could be helpful to a vet.