If by promotion of the 3rd night free you mean to make it evident for potential guests to see and possibly choose your listing due to the promotion, there are a couple of problems with that approach. One is that if someone is seeking 2 nights because that fits their travel plans, what incentive would they have to stay an extra night? The other is, that if you are willing for a guest to stay longer, would it not make sense to just reduce the nightly rate so it would show up in the search?
If you still want to do that, you could change the name of your listing to highlight the free night so it is visible. The process by which you would give this free night, is a refund after the booking is made. Now the Air BNB fee for the guest will be based on the 3 nights so the refund would have to take that into account as well. If there is occupancy tax based on the 3 nights, then that also has to be refunded. Pretty complicated.
