Without a secondary internet connection this isn’t possible.

The router is the connection - its the gateway (a term we don’t hear much these days).

You could setup an independent connection via a cell modem - becoming a secondary connection. This is common for remote locations or even small businesses that need a failaover just for management.

You could even have it on a single machine and have a vpn there. Then you could RDP/VNC to that one machine and manage things from there.

But there’s not much I could see you doing if the gateway is down anyway.