By accident I have developed some C-41 films at lower temperature (it was around 34-35°C) and the color cast of the shots is almost impossible to treat after scanning - and I adjusted the time, but only to slightly more around 4mins.
When developing sheet film in the HP CombiPlan tank, I have also resorted to prefilling the tank in the dark, as it takes some 30seconds to fill the tank with the lid closed.
I am even able to pour out the developer out of the tank, in the dark grope for the blix, pour in the blix, close the lid and start some agitation.
As for the drop in temperature, it's important to preheat the tank itself (let it lie open and submerged in the tempering bath for at least 5 minutes). Also I keep the tempering bath at 40°C, when I pour the developer in and start the agitation, the temperature is around 39°C, after the 3:15 development, the temperature is not much lower than 37.5°C...