You could use the majority of software that handles barrel distortion, and just either setup a profile and save/reuse that profile, or reuse the settings by syncing the settings. An example of this software would be PTLens, but many solutions exist.
Since this is a fixed focal length lens, you don't need to worry about changing the settings for each image - they will stay constant across all images taken with the same GoPro.
To get the initial profile setup, you could easily shoot a test chart or a subject like a brick wall to determine the correct settings. I don't see any issue with relying on the human eye for this, it is quite easy.
We have many other questions on this site about barrel distortion(and vignetting which may interest you), take a look at this search: https://photo.stackexchange.com/search?q=barrel+distortion