I finally managed to figure it out. The reason why the above config files don't have any effect is simple, but not obvious: There are two variants of the Moonlight front-end: moonlight-embedded and moonlight-qt. The files in /recalbox/share/system/configs/moonlight are for moonlight-embedded, but Recalbox uses moonlight-qt on RPi4. Therefore, another config file is used which is located at: /recalbox/share/system/.config/Moonlight\ Game\ Streaming\ Project/Moonlight.conf.
It was not clear to me how to modify the config to switch to 1080p. The file is normally managed through the Qt front-end and not directly edited. So I installed moonlight-qt onto another Linux box, applied the settings through the UI and checked the config file. There required settings are part of the [General] section:
[General]
width=1920
height=1080
fps=60
Porting the settings over to my RPi4 Recalbox finally made it stream my 1080p desktop in 1080p! Looks much better now!
