Hi Oliver, thank you for using our forum, my name is Luis I am part of the Small business Support community. The difference between Gateway mode and Router mode is, when the router is hosting your network´s connection to the Internet. This is the default setting, and Router mode when the router is in a network sharing routes with other routers, and another router acts as the network gateway to the Internet.
So in your case should be set it as Route Mode.
Now, about your issue, when you go to Port Statistics, are you able to see the DNS from each both Internet connections?
If you are able to see you could set an IP statically to a device just to test if you get Internet connection.
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