Transmission does not want to engage on cold mornings

2004 3 series BMW
When the temperature drops below around 40F, my daughter’s car does not want to move in forward or reverse for the first few minutes. I watched her rev it up to 5000 RPM to get it to move the other morning. I explained how bad that is for the engine and asked her to please be patient and let the car idle for a few minutes to warm up the transmission before starting out.

Two or three blocks down the street, after it has warmed a bit, the transmission engages briskly at the touch of the throttle and shifts normally with no unusual slippage.

I first noticed this after the last transmission fluid change. I suspected the wrong fluid had been put in it, and took it back to the BMW specialty shop that changed the fluid. The shop assured me that it contains the correct fluid and tried to tell me that there is nothing wrong with the transmission because the transmission computer is not storing any codes, and it worked perfectly when they drove it. Obviously 5000 RPM to move on a cold morning is not right, but I am probably not going to get any help from that shop.

Should I change the fluid again and see if it helps? I am hesitant because that full synthetic transmission fluid is expensive. Could a temperature sensor that says it is warm when the transmission is actually cold cause this problem?

Arrange to leave the car at the shop overnight so they can see the problem in the morning.

+1 to @BillRussell . If the car is warm they will not see the problem. The seals on the transmission are probably dry and hard, when they are cold the do not seal well and do not allow the pressure to build up, this is 11 years old. Once they warm up they swell a bit and seal better. The pressure builds up and the transmission works OK then. Give the shop a chance to look at it first, do not be surprised at the price to replace all the seals. You could also try any of the miracule cures in a bottle for fixing transmissions. These are solvents and other magic stuff that gets into the seals ans softens them up. Sometimes they work, sometimes they do not.