Thank you very much, I was able to debug now.
I investigated the underlying problem, and found out that in wordpress there is no way to access all custom post types across the network. This makes it a bit more difficult, and it looks like there is no way to bypass this either.
I believe the best and easiest solution would be:
– Create 2 search bars – one for the shop, one for the downloads
– Configure the first on the shop multisite
– Configure the second on the site where the downloads are enabled
In that case, the desired post types should be available.