Set specific interface for internet access

Solution 1:

It sounds like you're looking for a way to set the default gateway. You can see the current settings by running ip route. You probably want yours to look like something like this:

default via 192.168.0.1 dev eth0 ...
192.168.0.0/24 dev eth0 ...
192.168.1.0/24 dev eth1 ...
192.168.2.0/24 dev eth2 ...

These routes are automatically created by NetworkManager. Try checking Edit ▸ IPv4 Settings ▸ Routes ▸ Use this connection only for resources on its network on the two LAN connections to prevent them from becoming the default gateway.

You shouldn't need to work around NetorkManager.