Match moving and UV Project modifier with a strongly distorting lens

I am trying to follow the steps explained in

on a video which was recorded with a strongly distorting lens (GoPro Hero 3).
This is how the original footage looks like

Now if I put a plane as indicated below, and use the UV Project modifier (using the same movie clip as input), I get this mismatch:

My suspicion was that it is due to the strong distortion of the lens; and indeed, if I uncheck the ‘Render undistorted’ checkbox in the ‘Background Images’ settings, I get a perfect match between the original footage and the projected texture:

But unfortunately the track points and my plane is off from where it should be with respect to the background video.

What is the correct way to solve this problem? Is there a possibility to use an undistorted version of the video in the UV Project modifier, using the same camera distortion parameters which were determined from the tracking (without rendering it out)? Or what else?

Thank you