I'd also suggest that you might have the best luck just asking the front desk about it when you check-in. Explain that the later nights are from the free night certificate and it wouldn't let you book your desired room type with the certificate. If the nicer room type is unsold (especially at a limited service HH) you've got a decent chance (though certainly not guaranteed) of being able to stay in that room for the additional nights.

If we were talking about a week of additional nights, or a full service property where there's likely hundreds of dollars difference in the room rate then chance goes down much closer to 0 for a non-glob. But in this case, my hunch is 50/50.