The best solution i found for me and my skill level with pieces played and situation is to practice being able to play without warm at all.
I gave up warming up or having a need and desire to "play" before playing. You will not always have the luxury of playing warm up and palying through a work before a performance as you found out.
I changed my practice approach so when i go in to practice and begin playing i meediately begin playing and do not do a warm up. I routinley make it a point at random times of the day between tasks to pop in at piano, play through, and leave.
Same with my piano lesson i go and i sit and i play, sometimes my teacher offers to let me warm up if i
like, i decline.
It doesnt always go well but that is the point, mess up while and where it is safer to.
There have been countless recital, jury, johnny on the hot spot performance opportunities, venue outside or with blasting AC or no Heater etc where i maybe waited 40 to 50 minutes for my turn to play or like with my lesson where i drive > 1 hour one way to reach my professor studio, warm up is unrealistic expectation.
I found over time less and less reliance on it and my playing from "cold" got way Way WAY better so now its almost no sweat to play cold. I am still working to imorove this.
It is also a function of your prep and how well you know the work, the more solidly learned the less warm up needed.
If you always practice by doing a warm up, you will almost all but certainly need a warm up, you have conditioned your nervous system and psyche to rely on that.
It is not easy but it is a soft skill worth developing.
Then it is also good to have a pretty and soft very short work in your back pocket for those rare lovely instances you have the freedom to play more than what you planned on
Occassoonally if you have to play on a instrument dramatically different than what you normally sit at, it can help to play the work not as a warm up but to get a quick feel for action, pedal, voicing, volume etc of the piano, info you need in order to make all sorts of tiny addjustments. The skill should eventually be developed you can do it cold in first few notes of your performance work, but it helps to have a "prelude" ready for when that is an option given to you.