Thank you for your reply!
No, I haven’t tried that, but I resolved my issue anyway:
Apparently, the hosting provider was a bit strange in this regard (hostpoint.ch). I tried several settings, including pointing childdomain.com to the IP address of the multisite, using a CNAME record and even with an HTTP redirect.
Nothing worked, because hostpoint.ch requires the creation of “websites” (records) that sort of create a connection between a domain and an actual directory on the server. So if you just point a domain to an IP address or directory without creating such a “website”, it won’t work no matter what. I think that’s pretty strange but in the end we got it to work.
Cheers!