I’d like to show the travel status (confirmed, almost full, sold out, etc) in the status column of the booking table for each pricing option. I noticed that booked pax is not increasing if a booking and it’s payment are confirmed.
Regarding your query related to travel status, for now only booked, pending, canceled, etc status can be displayed for booking status column in the backend section(booking table). However, you can use the sold out message option by going to Admin Panel > Trips > Individual Trip(edited) > Inventory Option to display the message for a single trip as per your requirement under the booking tab option in frontend section. You will get the option like the screenshot provided below if you go to the individual trip and select the inventory option.
Also, the inventory option gives you a feature to provide the information regarding the seat left and booked seat. You can find the feature in the WP Travel utilities and in the WP Travel Pro. These plugins offer you the feature you mentioned.
You can refer to the documentation to be more clear on the features,