Hello, this is my first comment here, I am not an expert and this is not my native tongue.
I had the same problem for a long time and finally could find a homemade solution.
To sum up, you need to have an app to browse your device's files so that you can move the mp3 to the folder that your player app is using.
The podcasts' app I use is 'ivoox', and the files browser app I use is 'File Manager'. My player is 'google music'.
If you open ivoox and go to settings you can see the route where podcasts are being downloaded. Note it and then go to such route by using 'File Manager' and once there cut the mp3 file and paste it in the folder where your music player is storing the files. In my device, google music player uses this route: storage\emulated\0\Music, I think is the same route for all devices.
Now you can find your podcast in Google Player Music and syncronize it as usual to your SonySmartWatch 3.
I hope you can fully enjoy your SW3 now when running!