The only option available for this contract is "commuting" there is NO option to reside in Japan with this contract (no option that will be sponsored by the company that is, they WILL NOT arrange resident status in Japan) since you will not have official residency in Japan, your salary is not subjected to Japanese taxes...., remember that you will not actually work for AJX, you will be employed by the contract agency....! Your salary will be net and you are responsible for applicable taxes on your country of residence...!
One more thing, CREW is a US based company....., if you select them or not as your contract agency I think it is a good measure to speak to a tax attorney to get a clear picture of your tax liabilities under this type of employment..., it could be quite complicated depending on your own country tax laws....!