On the HomeRunner confirmation page, trigger your purchase conversion tag on the custom event named purchase. The HomeRunner plugin pushes this event into the dataLayer when your PMS returns the reservation as confirmed.
Why not a page-URL trigger. A page-URL trigger fires before the purchase push lands. As a result, the order ID and value arrive empty, page refreshes cannot be deduped, and inquiry confirmations are counted as purchases.
Lead tags. When the booking is not yet confirmed, use generate_lead as the trigger for your lead tags.