The 'Sent' mbox file does not have an extension. It is a file called 'Sent'. The 'Sent.msf' file is just an index file used by thunderbird.

You need to get a copy of the 'Sent' file and put it on 'Desktop. Then rename it to something like 'OldSent' no extension - then right click and select 'Copy' Exit Thunderbird Access profile folders. In 'Mail' folder you will see 'Local Folders' folder. Paste the copied 'OldSent' into the 'Local Folders' folder.

Start Thunderbird. Old sent emails should now be visible in 'Local Folders' mail account. You now have the option - either leave them where they are like an archive after all you can access them or put copies back on to server but it will use part of your quota.

To put back on server - highlight a small batch of 'OldSent' emails - use right click 'Copy to' and select the imap Sent folder. I advise 'Copy to' because if something goes wrong you have not lost the original.

Please Note: You cannot simply copy paste the 'Sent' mbox (no extension) file into the imap account. The first action that occurs when you start Thunderbird is imap account will check server and display what is on server. So as Sent is empty on server - that is what will occur in Thunderbird, so losing the file you just copied over. Hence why I advised you to put it into the 'Local Folders' mail account. Then you have the option on leaving as it is or uploading batches of copied emails back onto server.