I had the same problem and just found the most easy solution!
In my case, the problem was that I tried to install the Multisite with subdomains first, so I created the subdomain already at my hoster. If you change your mind afterwards like I did (installed a new wordpress, this time with subdirectories), you have to DELETE the subdomain at your hoster!
No more 403 for me after simply deleting my empty subdomain folder. Hope this helps somebody!
