If the Night's Watch is a client state and not a vassal of the monarchs of Westeros, doesn't that undermine the argument that the Ides of Marsh was justified? Even within a client state, the head of state / government still generally has latitude to conduct foreign policy, and it takes explicit direction from the patron / imperial state to countermand, undermine, or replace the leader.
I don’t think “client state” is quite right - for example, the King on the Iron Throne cannot replace the leader of the Night’s Watch; only an election by the brothers of the Watch can make a Lord Commander.
And the Ides of March was never justified on the grounds that Jon Snow was betraying the King on the Iron Throne - Marsh et al. were justifying their actions on the grounds that Jon Snow had violated the internal customs and taboos of the Night’s Watch.