So, I wonder if my genetic disease is possible or a total fiction
Its hard to say, because itching isn't perfectly understood. There are a few different classes of nerve fibre involved, and it seems to be part of the pain reception system, but full understanding is lacking. It is possible that you could only become truly insensitive to itching if you were also insensitive to pain, and I'm assuming that isn't something you really want.
There's a possibility that there are itch-inducing proteins and receptors on nerve cells that are triggered by them. Whether it is possible to malform either or both of those and suppress or reduce the feeling of itching isn't clear, but there is an avenue of exploration there. Chances are that modifications to the genes which ultimately encode those proteins will also have a bunch of other effects too, though. Few genes are convenient on/off switches for a single system and nothing else.
It might be possible to get something similar to being itching insensitive by having some change or damage to the anterolateral system. This system, amongst other things, deals with identifying where painful stimuli are originating and triggering basic responses, like "put down the hot thing" or "scratch the itchy bit". It is possible to conceive of someone who could feel an itchy sensation and identify it as itchy, but not actually care about it at all and have no scratching responses.
What would be congenital insensitivity to itch's mode of inheritance
Because the exact nature of itching isn't very clear, there's no obvious way to disable it without disabling a bunch of other things and as such the nature of the inheritance pattern associated with genetic insensitivity to itching can't be defined.
Pick one you like, that suits your story, and that'll have to do.