One way I might try to do this would be to use Time Remapping to change the timing of the lumberjack’s swing.
First I’d add layer markers for each beat on the layer that has the music track. To do that, move the CTI (current time indicator) to the beginning of the comp, then with that layer selected, hit zero on the numbers pad to preview just the audio, then hit the * key on the numbers pad to add a marker as you hear each beat. It won’t need to be perfect, you just need a visual to time things up in the next step.
On your lumberjack layer, enable Time Remapping (right-click layer and choose Time > Enable Time Remapping). This will add a new Time Remapping property to the layer with two keyframes, one at the beginning of the clip and one at the end. What you’ll need to do is scrub thru the footage and add a new keyframe at every swing.
Once you have the swing keyframed, you can adjust the timing of each keyframe to match a beat marker on the music layer. Preview it and make timing adjustments as needed.