Why not just use Fastmail instead of Inbox.com? You can change the generating system target address so that the messages are directly sent to a Fastmail alias for new messages. Then import the existing messages to your Fastmail account via a big IMAP transfer:
https://www.fastmail.com/help/receive/migratemail.html
Since you are importing the messages using IMAP (similar to an IMAP client), they should be in correct format rather than using forwarding headers.
Bill