A cooler on an s23 ultra will gain you at most 2 degrees and are a noticeable drain on your battery, the extra battery charge cycles defeats the idea that a small heat reduction will save your phone's battery.

There's a reason no phone coolers show actual benchmarks for their cooling performance.

Turn down your emulation resolution scaling if you think you need to save heat. The 8gen2 can do the most demanding emulation at 40-60% usage. Nothing pushes it to the limits outside of dedicated prestige android games like genshin impact.

The best space saving controller I have is the kishi 1 that I keep in a case to protect it in my bag. I 3d printed an insert that fits my case. Search for s22 ultra since it's been around longer and exactly the same size (see yeggi or thingiverse).

https://www.amazon.com/Razer-Controller-Gamepad-Android-Renewed/dp/B08VW7KGV2/

https://www.amazon.com/Hermitshell-Travel-Razer-Mobile-Controller-Android/dp/B09FXD71MQ/