The best way to approach such questions is to remind yourself that you do not have one client who has a wife.  In fact, you are preparing a joint return and you have two clients.  Does Client #1 (the husband) qualify?  Yes, because he is a full-year resident alien.  Does Client #2 (the wife ) qualify?  Yes, because her "filing status is married filing jointly and her spouse is a U.S. citizen or resident alien and she chooses to be treated as a U.S. resident."

However, since the children were not in the US for six months, the taxpayers only qualify for the EIC without children.  The children do qualify for the child tax credit, since they (presumably) lived with Client #2, their mother, all year.  They may be considered to have lived with Client #1, their father, all year also, because of "temporary absence," but we don't know the details.