I think maybe you guys might misunderstand. The only reason why any government exists in an official capacity is because it has the support of the people and military. Internationally, other parties will have their own ideas. If such a govenment is to rise up and be recognised by its peers as such, then the controling force that means to control it is then open to scrutiny and sanctions. This happened in South Africa, Tibet and also in Ireland when international recognition was sought.
Trade embargos go a long way. A bully can start to think twice when others refuse to trade with them out of principle. The Alliance does not need to go to war in order to support Lugh. It just needs to change the way it deals with the Federation...
Should the Empire, Alliance and other free worlds choose not to recognise LFE as a legitimate Government and impose sancions upon them while recognising CSG instead, that goes a long way to making Lugh a very unprofitable entity for the Federation and highlights the illegitimacy of their claim to control it.