I don’t think you need to be UNDER the car to see the motor flopping from side to side, just watch the doghouse!
If the linkage is stock setup, check the rod that goes from the trans bell crank to the doghouse bell crank and runs through a hole in the front tin. It is unusual for there to be enough movement for the rod to pull or be disturbed much by movement but it could be hitting the tin where it passes through the hole.
The cam you have is a point of interest to me. The cam is the “brain” in a 912 motor and it will profoundly influence the behavior of the motor. What cam grind do you have? Do you know the timing duration and lift of your “3/4” cam? If you have a stock to mild 280 to 288 deg (IMHO 288 deg max grind to have in a street driven 912 if you want it to be drivable) cam use the 031, 022 or 050 dis and set about 4 deg initial advance. The final advance should hit around 2800 to 3000 RPM and 32 to 34 total depending on the distributor used. This is average setup, the correct timing also depends on the compression, gas and cam used. If you have 300+ deg cam you can use the 009, but you will have to set in about 10 to 12 initial advance and forget low-end drivability and torque. You will end up with 32 or so final advance. Watch the motor temps! Too much advance will cook a motor the same as if it was running lean. If you set the 009 for a good final advance for top end power it will be a hard starting pig and have drivability problems at the bottom end. Even then, the final will occur too early for street use. Just another opinion but hot cams suck in full weight street cars.
Cams over 300 deg will trash bottom end power and low rpm drivability but give top end and keep the power going past 6000 rpm. (Stock cam 912 motor will go flat at about 5800 RPM) If you have a 290+ deg cam that very well might be the center of your problem. I use a 288 in my semi track motor and it drives well street and track but only after I spend some time setting up the carbs for low rpm use. Mild cam, 9.5 to 1 and 1720 cc but has dyno well over 100 HP to the ground with a nice wide torque band. When I track the car I have to change the jets and the timing but it pulls well to 6800 RPM and the power cams in around 4000 RPM.
There is a good book on 912 tuning for street and track comming out this fall. I have seen some of the stuff in it and it looks great.