Yes Widgets can’t be replicated and the GameMode only exist on the server so you need another replicated Actor to execute a RunOnServer Event.
The PlayerController or a Possessed Pawn can be used for this or any other replicated actor spawned by the server owned by the client PlayerController.
When the button is pressed you use the PlayerController/Pawn/ClientOwnedReplicatedActor to execute the event on the server and that should be it.