Seems unlikely that it's anything to do with the LAN, assuming you can still reach the router. (If you're losing the connection to the router, then it's not a 4G failover problem.)
The answer (or at least a hint of what might be going on) lies in the router logs, if you have access to them. Could be all sorts of things. Off the top of my head:
- Router fails to detect a drop in service, for whatever reason (eg. the DSL is up but no data is flowing, or an upstream routing problem means you can only see part of the internet, but whatever's being monitored still works).
- Dongle doesn't work, for whatever reason (buggy drivers, hardware failure, etc.)
- Dongle works but LTE fails to connect, for whatever reason (eg. LTE provider also has issues.)
- Router fails to switch between DSL and LTE in a smooth and timely manner.
- Some weird edge-case effect that you'll only believe once you fully understand it.
In my experience most non-trivial network problems come into the latter category.