Hi Ymdemar,
The job is clearly a commuting contract and AJX will not assist in any way in obtaining residential visas. During initial training you will be provided with a Visitors Visa (or Alien Registration Card) but once training is complete you will hand that back and the come and go from Japan just as any other international pilots would do on a Shore Pass, which is valid for 7 days.
There are a few guys here who are married to Japanese girls and obviously they can reside in Japan and their kids can go to the schools etc. There are also one or two pilots who split their days off into smaller blocks (i.e. less than the 7 day limit of the Shore Pass) and have these days off in Japan. Effectively they are living in Japan but they are not residents, if that makes sense. However, the ones I'm thinking of have no kids in the country and obviously do not use the Japanese infrastructure of schools, government departments etc.
So, in short, you cannot get a Residents' Visa through AJX to live in Japan and move your family there. It may be possible to arrange your situation to stay there during your time off but I would imagine you'd then have to pay fairly large fees for schools etc. I don't know this for sure but no doubt someone else will correct me if I've got it wrong.
If it helps at all, we have 3 (?) Italian pilots who commute back and forth to Italy and I have not heard of them having any problems with the commuting side of things.