I think a budget android handheld could work well for game streaming. TechDweeb and Retro Game Corps newest videos show android handhelds that cost less than 100€.

I think even 1cm would be enough. Take a look at the ifixit steamdeck heatsink replacement guide to see how small the heatsink is.

https://www.ifixit.com/Guide/Steam+Deck+Heatsink+Replacement/148904

If you go to chatcontrol.eu you will find more info about chat control and how to fight against it.