A few points.
This will take some years to develop.
We don't "have to" fight at all, we can pop popcorn.
By "underdog nation" I refer to the commonly occurring trait of all Western European nations to attempt to tyrannically dominate their neighbors (and their own populations).
Muslim agitation could be ONE of the reasons a population may come into serious conflict with the "New EU". Say French citizens finally tire of Muslim rioting, assaulting, raping and weekly arson. They implement harsh policing methods, quick trials locally, and draconian punishment. The Moos squall to the New EU authorities charging racism and religious persecution. The New EU sees an opportunity to cripple France's economy and competitiveness and collect monetary fines and penalties, sides with the Moos granting them religious exemption from French law and imposes Sharia in Moo neighborhoods. They also send men with guns to insure French bureaucratic compliance. Most speak German and wear their New EU silver badges on heavy silver chains around their necks. A gunfight occurs between French local police and New EU police when the locals attempt to arrest a Moo murder suspect. The ICC demands the French locals stand trial, the EU's get a pass. The French president orders all EU out of France, the EC moves military units into France and demands more from other EU states to "put down" the French, and IT'S ON.
Admittedly, far fetched, French authorities would sell every native born Frenchman and all their police officers down the river for the bucks of EU commerce into their Swiss accounts. But a Bosnia style civil war is much closer to a possibility. In any case at this point the US will be asked to intervene, what we do depends only on us.
Hey Sarkozy, are you listening?