Tried again with the updated version of Toolkit II (I also removed any files that were left after unsubscribing from it on steam, before resubscribing again).
Result was the same. As soon as a player/client tries to connect to the lobby, they will be booted back to the main menu with the missing dependency instead of automatically downloading it.

I saw Larian is preparing for patch 4 and it will include improvements to mod sync issues. Don't know, if this is that issue they might be fixing, but here's hoping that it will. :P