Doesn't sound like transmission failure to me. I had exactly the same issue to the letter on my x5. You need to look at what your live data is saying for the fuel trims at idle and revving at 2k ish.
Without load mine revved perfect and Idle was perfect. Under load it would do exactly the same as you describe word for word. I had no codes at all, I assumed vac leak or faulty MAF. Finally yesterday after several major strips and investigations I had a code pop up, low maf circuit reading. I had never had this before while having this problem for months. Fuel trims were well out, replaced the maf for a new Bosch one and bingo, car is like new and goes like a train with no faults at all...
I had all the shudders and coming to a stop was identical to your issues, it really did feel like the transmission was struggling as you described.