Thank you for the file.
Your APIM is GB5T-14G371-CFC, which is not MY20 according to our SYNC 3 APIM Hardware Database - CyanLabs . Remember that the log can’t see the screen part number, only the APIM…
Now, about your current situation, I would not go to 3.4 right away.
Your aim should be to make CP work on your current build. I’m also running 3.0.20204 and both Android Auto and Car Play work.
I’m saying this because if your issue is because Ford made you bypass an interim version that enables CP and AA, there’s a chance that going to 3.4 won’t fix it and you would be stuck.
Review the following thread, specially this message → 2016 Mustang Sync3 1.0 updated to 3.0 and no Apple Carplay! - #10 by SaNdMaN …
You could try CP while being on 2.2.17xxx as well…