You can use tvQuickActions. It allows change input in two clicks. In next version will be available in one click. (sorry for ads, but it is the solution)

Remapping can work but there is a setting that automatically switches to the last used HDMI input on boot.

Try this http command on a web browser (e.g from your phone), and see if it works

http://DEVICE_IP:6095/controller?action=changesource&source=HDMI1

DEVICE_IP is your TV IP Address.

source

EDIT: if it works, you can then map that command using Button Mapper

Doesn't it support the HDMI-CEC? That way you could control your Mi TV via your original TV remote. You need to enable it in the settings though. Your TV must support it.