Continuing to list as second affiliation an institute where you've never had a paid position
It’s dishonest (and therefore unethical in almost all normal circumstances) to represent yourself as something you’re not. That’s what your grant employee wants to do by listing his affiliation as a place where he doesn’t hold an official title (and it’s irrelevant whether “someone said it’s okay” unless perhaps that someone is a dean or a similar high-ranking person who has the authority to give people official titles; it’s also not strictly speaking relevant whether he had a “paid position” or not - some official titles are unpaid).
As for what to do, I agree with your reasoning that this is your responsibility since he is paid with your grant funds. It’s perfectly appropriate for you to say that he cannot write this dishonest affiliation in any paper that is funded by your grant. If he does not accept this, you should check with your institution what enforcement or disciplinary measures can be taken in connection with this type of employee insubordination. It may also be worth trying to talk to him informally (perhaps in the presence of a “neutral” senior colleague) and see if you can understand the motive for this behavior; possibly it’s based on some misconception on the part of the employee, and you may be able to convince him that there’s nothing to be gained, and much to be lost, from this type of dishonest behavior.