How did you setup your account in K9?
I guess for sending it should be irrelevant if you use IMAP or POP as it uses SMTP for sending.
But maybe the folders are not set up correctly in K9. Check Settings / account / folders if “Sent folder” has set the correct remote folder
It works for me, using K9 6.7 beta and Thunderbird 102.11.0. I just tried sending from both, and sent mail shows up immediately in both Sent folders. I’ve seen to need to manually sync, everything just shows up in both applications.