Most of what you’re seeing is probably the functions in CharacterMovementComponent that deal with client/server position error correction, such as:
ServerMove
ClientAckGoodMove
These are sent at a high rate to reduce latency when movement occurs. You bring up an interesting case though, when players aren’t moving they probably don’t need to send it at such a rate (we already throttle it, but it could be throttled more I suppose).
Replication of variables only happens when the values actually change, so when objects aren’t moving or rotating then network traffic should be low.
I’m curious about your setup, because it sounds like you should don’t actually need a local player character (which would remove the network calls I mentioned above). You could get away with something more like the Puzzle template that only uses a PlayerController and no local pawn. All pawns would be on the server, and the local PC would send RPCs (from mouse events) indicating which pawns they want to move. Those pawns would move on the server and replication would occur when they move, but traffic would be low when they don’t. The pawns can still be characters-- the character traffic is only high when there is one possessed on clients.