Innius connects my machine to the internet, and the same machine also has a LAN connection on my local network. How do I know I am not introducing a security weakness into my LAN?

Innius only needs outbound HTTPS connections from your local network to the internet. Your firewall settings control which port(s) are opened up for outbound HTTPS connections. In that respect, innius requires similar settings, because an IT department will typically allow for web browsing from within the LAN.

Innius only requires an outbound connection to one specific IP address, and no inbound connections at all.