The car needs to be in the air, and then you need to look at the floor where the subframe mounts to. If you hit the area with a degreaser like brake cleaner, many times it will run through the crack...or because you got rid of the road grime will have a better time seeing the crack.
I've welded plates to many e46 floors for cracks, the mount will look fine. but the flex actually comes from the floor. e36's and e46's have similar design flaws on the rear floor.
There's another test you can do where you load up the wheels and using a helper you can actually see where the floor gives.
Check youtube. there are many videos on how to check for cracked floors.
I'd check there first. Sounds like it happens when your putting load through the drivetrain. (first gear launch) Also your motor mounts being busted won't help the situation. But with busted motor mounts you'll hear more of a rattle under load or around a corner. If your mounts have been busted for a while it will also put extra stress on the rear floor.
Also when you say jerking back and forth....does the car buck like your riding a mechanical bull? or is it less obvious