I live on the Miura peninsular not too far from William Adams fiefdom. It’s possible to walk up the hill from Anjinzuka Station (The navigator’s hill) and see his and his wife’s grave.
Civil registration started in England only in 1837. Modern koseki system in Japan started in 1872, but different systems of civil registration existed way before that, including one created under the shogunate (though many years after Adams's passing).
And indeed, they couldn't possibly careless about his married status regarding him marrying a Japanese woman. His declaration of death was so that he would be free to serve the shogunate freely, without the obligation of return to his family in England.
Neither bureaucracy nor family honour.
I mean, bear in mind that about 30 years before Adams was born, Henry VIII had broken with Rome so that he could get a divorce. Religious views on the sanctity of marriage in England were, ah, evolving rapidly at this point.
As far as I know, Henry VIII never divorced any of his wives, they all ended because of annulment or death.
(Not that this was exactly unprecedented; the whole thing kind of came to a head because the Pope wouldn't grant a spurious annulment to cover a divorce to Henry, but various Popes had certainly granted dodgy annulments to various monarchs _before_.)