Gmail automatically copies sent messages to the Sent Mail folder, which you should subscribe to in TB. If the phone mail app copies to the same folder, you should be able to make a filter that copies from that folder to Local Folders, but note that filters only run automatically on the Inbox. For any other folder, a filter has to be run manually. Post a picture of the filter you've tried.

Hmm I feared that might be the case that only the Inbox could have automatic filters. I sometimes use TB to send emails sometimes my phone so I couldn't imagine how to set up a manual filter that didn't copy all the sent mail after a certain date for example rather than just the ones that were sent by phone.

If your phone copies sent messages to Sent Mail, and you want those messages copied to a folder in TB, you could add a unique string to the Subject of messages sent from the phone, e.g. sentbyphone, and then in TB create a filter that runs manually on Sent Mail and filters messages with the Subject containing sentbyphone and Date is after a certain date.

That's a great idea, thanks! Could it be done with tags maybe?

I don't think gmail supports keywords, so tags applied on one device aren't visible on others with the same IMAP account.