Thank you for your answer again. In the end, I found an answer while digging though the Inspector on the website. In case anyone struggles with this problem in the future:
You have to add
.w-background-video>video {
max-width: -webkit-fill-available;
to your custom Head code in the project settings inside the style tags.
I tested it with Chrome and Safari on Mac, and Safari on iPhone – it seems to be working on these browsers! 