For anyone else out there who has trouble with keeping rhythm and is tearing their hair out during the "double beats" section/round 4, I would recommend focusing on the button press AFTER the double beat. I spent a good 30 minutes thinking I was screwing up the timing on the double press when really I was just missing the beat after the double.
It helped me to think of the double beat and the one after as three presses in a continuous rhythm, which is quite quick! be ready to move your finger to the other button pretty fast. Still a fairly challenging achievement, but hopefully this helps someone.
EDIT: GamerWedders has posted some great tips in the comments below, so I'm going to quote them here
Some additional tips:
* In case people haven't noticed, the bonus round where there are 8 buttons per set rather than 4 is actually just doubled up, i.e it's the same 4 buttons repeated. I was a bit frazzled by the time I reached this so I didn't pick up on it right away.
* You can spam the buttons. This gets a bit hazardous in the faster rounds, but I found it useful to just start spamming the first button as Stubbs walks to the dancefloor, and start spamming the next button each time I heard the 'ackowledgement beep'.
Good luck folks.
I would also like to add that you can pause at any time if you are feeling overwhelmed or stressed. If you missed a part of the sequence you need to remember, you could even record a clip while paused and watch it back until you feel confident on what you need to do.