I hope you haven't gotten rid of your M3 yet. I stumbled across your thread a while back ago when trying to find a solution to the jerky takeoff problem I've been having with my 03' 330i. It was doing the same exact thing your M3 was doing in your video.
I found there was an O-ring missing on the air filter housing where the MAF sensor clamps to it. After replacing this O-ring, my car can now squeal tires off the line without bucking like a wild mustang.
Check to see if this O-ring is missing, or if the rubber intake boot that clamps to the throttle body is cracked.