Deactivating your Hostaway Booking Website does not affect your HomeRunner website. Your HomeRunner website is independent of Hostaway’s hosted "Booking Website" product.
Why this is safe. HomeRunner connects to Hostaway through the Hostaway API. That connection keeps your listings, photos, pricing, calendar availability, and reservations in sync, and the Booking Website settings do not affect it.
What you can do. You can toggle off either domain in Hostaway under Settings > Domain & Security, or deactivate the Hostaway Booking Website altogether. Your HomeRunner site continues working normally.
Important caveat. Deactivate only the Booking Website product. Do not close or cancel your Hostaway account itself, because that is the system your HomeRunner website syncs from.