Improve how lobby creator can change lobby settings

[evanpelle]

Instead of lobby creator using the PUT /game/id endpoint, it should update the game config via websocket. The lobby creator is already authenticated in the game server, so using websocket to update config is more simple.

[evanpelle]

also start game can be passed via websocket

[VariableVince]

Would a good implementation also fix #344? @ryanbarlow97

[VahantSharma]

Makes sense.

@evanpelle The linked PR (#2761) is intentionally scoped to the original goal of this issue: moving lobby configuration updates from the REST endpoint to a WebSocket-based flow, preserving existing behavior and validation.

Additional improvements like sending “start game” via WebSocket or addressing API permission gaps (#344) feel like natural follow-ups, but orthogonal to this change. Keeping them separate should make review and rollout simpler.

Happy to help with follow-up work.