Liberty military coup actually been a thing, in vanilla campaign. Could it happen again? Sure, but not because of just Rheinland. Maybe some more apocalyptic scenario in relation to Nomads/Gallia
The President is an extremely visible national figure, so it's unlikely that his absence would not be noted. It was silly enough that it happened in the Vanilla Campaign, so let's not play that card twice. The Libertonian Government may have fabricated reasons for the war with Rheinland to appease war lobbyists and parties who wanted to assert the house's dominance over Sirius (Keep in mind their war started before Gallia was on scene). With the current situation, a full mobilization is likely to occur in the near future when the events of Royal Flush play out fully, and martial law may be declared in some capacity.