Hi Furqan,
One Customer application to many appointments, so Appointment and Customer Application have N:1 relationship, you want to get value from child records to populate parent record.
Relationship mapping is really not feasible, and you mentioned that you don't want to use local storage.
From your screenshot, Customer lookup is not required on the quick create form, so you can create Customer Application without populating customer lookup firstly.
Then you can use workflow/flow to populate it automatically after saving the customer application record
Because that customer application lookup on appointment will be populated after saving the customer application record, then you can update the record.
