Hi Mike,
Only the first two Contacts for a Customer will sync with QuickBooks Desktop. These are designated by the Contact Type field in Method, with Contact indicating main contact for a customer and AltContact indicating the alternate contact.
The reason the field is shown as read-only on the Add / Edit Contacts screen is because it's not something that should be changed. Because QuickBooks handles the relationship between Customers and Contacts a bit differently from Method, changing this would result in synchronization conflits. If you have to change the Main or Alt contact for a customer, it's recommended that you manually swap the data between the two Contacts (or create a screen which automates this process).
Hope this helps.
- Justin