Which makes me wonder how much trouble it is worth taking with the fine control of the temperature for a single event - perhaps just once or twice per day?
I think I see what you mean. The OPs requirement to reach temperature as fast as possible is not compatible with the requirement not to overshoot. So why not just open the valve, get to the target temperature then close the valve. Yes you will overshoot but you will know by how many degrees and how long it took to get there. That information can then be used to keep the temperature within a band of the target for the rest of the day.