Rent-a-center need one week leniency - can I keep the computer one week without upsetting them too badly?

Your best bet is to call them, explain the situation and see what they say. If they say bring it back, then do it, otherwise they can tank your credit score if you have one and it becomes a repo issue. You do not want those guys coming to see you. They'll take it back, then you'll have no computer and no way to get it back.