Solution to my problem:
In my case, I could either achieve
(1) - an internet connection of the device (Ping possible) but no internet connection of connected clients or
(2) - no internet connection of the device (no Ping, no RMS) but internet connection of connected clients.
As I found out, the problem is, that the Mesh client does not inherit the DNS-Settings of the gateway, although the Mesh gateway is registered as gateway by the Mesh client and - as I understood by support - it should actually inherit those settings. So, to get access to a DNS Server, a custom DNS Server has to be entered in the WAN interface. So, here the step by step approach:
- Set up Mesh gateway as in the configuration guide
- Start setting up Mesh client from factory defaults
- Set fix IP adress in the setup wizard.
- Set up Mesh SSID with correct SSID + password, everything else left at factory settings
- WAN-Interfaces: a standard WIFI0 interface is set during configuration of Mesh SSID
- WAN-Interface: WIFI0
6.1) General page: Protocol = DHCP
6.2) Advance page: Use default gateway = On
6.3) Advance page: Use custom DNS Server: 8.8.8.8 (google)