This is along shot ... back before synthetic fluids where "a thing", our cars (manual and auto trans) would have issues with cold. Soon as we swapped out the OEM fluid with the fancy-shmancy stuff they shifted just as good whether it was -20F or 100F. So maybe a fluid change can help? If the OEM fluid is synthetic maybe changing to a higher-quality brand can help? Or is this one or of many good reasons not to spend winters in MN ?
I specifically recall two mornings ... one in a 1980'ish Ford Escort, at 4am January in MN, I had to go to work at Hardee's to make the biscuits, and the stick just didn't move. After idling a few minutes I managed to get into first so I could at least get the car moving a little, which helped to warm it up.
Another frigid cold morning, after a night partying in Mankato, we got into my 1984 Camaro Berlinetta. The 305 4bbl fired right up, put it into Drive and .... no propulsion. Reverse? Same nothing. We sat there for 5-10 minute with our buckets (because hung over) thinking "now what" as the heater started to defrost us a little. Tried the transmission again and it slowly brought down the RPMs and we were able to get going.