I have a 4.4 (year 2003) that has a "shudder" in 3rd gear too & It sounds similar to your problem. In my case, the shudder occurs, as the gearbox is about to shift into 4th, on a light throttle. If I am driving more "enthusiastically" the shudder doesn't happen.

I have found that if I drive it in sports mode the the shudder never happens. I have come to the conclusion that the torque converter lock up solenoid is "having trouble" when in standard drive mode

I think the shift sequence is a little more basic when in sports mode & the solenoid is not used to "smooth the shift in the same way as it does in standard auto.

I am not sure what to do about my problem. Maybe a genuine BMW unit is the way forward. I worry about paying big money for a repair or a refurbished unit. At least a genuine exchange unit will have all the important bits brand new & a good guarantee to go with it.

Good luck with your problem.

Regards

Dave