In the light of the impossibility to retrieve a muxed .mp4, the userscript is no more useful. Alternative is to paste the url of the page to yt-dlp.
My wife needed to download more easily a bunch of videos from ARTE.tv for her students.
The script shows you right under the current video a list of downloadable resources. For each link you can :
- left : open the video in a new tab (normal behavior)
- right : copy the link to download the video in a downloader manager
The script gets its labels and other texts from the API, so they are already translated. I added a few translations based on the six languages available on the website. Feel free to tell me if they are wrong.
I developed the script with Tampermonkey 4.11.6114 and Mozilla Firefox 77.0.1.
Feel free to tell me or the others in 
It « should be » compatible with Greasemonkey and Violentmonkey.
Installation : just copy 
^ ^ ^ German translation ^ ^ ^
^ ^ ^ French translation ^ ^ ^
ARTE offers a desktop application named « EducArte offline player ». But it is slow and you have to go to the website first, add the video to your account, then (slowly) download the video with the application. Neither done nor to do. Yet, you are fully responsible of any usage you could do with the script which could be against ARTE policies.