The school says they will send someone to BKK to make the license/waiver but there is some "issue" with the licensing system and must wait 7 days.

Can I extend my current non-B visa for a week, then once have all the paperwork, go back and do the year extension?

That will depend upon the immigration office where you do it. You need to check with them what there policy is.

In the past I would of said no problem but I have seen a report that Bangkok told somebody no extension after the 7 days under some new policy they have,