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if the transmission is going into limp mode, the car will be recording relevant, current faults relating to the cause of the limp mode - this will be required for a full diagnostics. If the shop you tool the vehicle to said there were none, then they do not have the proper equipment to read transmission codes and were likely only reading old engine codes. A proper scan with a tool that can read transmission codes will be required to verify this issue.
given what you have described, I can suggest a likely cause:
the speedo going out indicates a problem with the vehicle's reading of road speed from the abs system. A road speed reading is a required input for the transmission to operate properly, so it goes into limp mode. This would be most likely caused by an abs system failure, or by a vehicle communication network issue elsewhere.
if you need a non dealer option for a proper diagnostic on the transmission and the abs system, check out www.bimmershops.com or www.bimrs.org for options
Please let me know if you have further questions. Please do not forget to click a positive rating as this is the only way I receive credit for my answer, thank you. Best wishes with your vehicle.