if i have a little time will do it....Hello @guigux,
Great work of testing so many Distro's. I would suggest you to try Manjaro Linux from manjaro.org.
I hope this will work out of the box as they provide most of the drivers support out of the box.
There is a Manjaro Hardware Detection Tool called MHWD which can help you find and install the needed driver itself.
I will wait for your response and try to help you get Manjaro running on both screens.