Gen Lee and the ANV would have made mince meat out of Napoleons Army. (Even if you upgrade the Frenchy's rifled muskets to Springfields )
So in effect you are saying that the army of Generals Grant and Meade was better than Napoleon’s army. General Lee failed in trying to make mince meat out of them.
Besides, after a Union defeat or accession to Southern independence, the Confederacy would likely have rapidly demobilized, so there would not have been an Army of Northern Virginia, nor a Union Army of the Potomac, for that matter. The only Confederate state with Indian problems in the 1860s was Texas, and the Rangers would have taken over the whole responsibility for the frontier, a task they shared with the U.S. Army from 1845 until 1860. So for all practical purposes, the Confederate government would have been toothless.
Ditto for the Union if the British enticed the New England states and upstate New York into joining the Dominion of Canada in order to protect the St. Lawrence Valley and the eastern Great Lakes. The New England merchant and fishing fleets would gain the protection of the Royal Navy and access to British Empire markets. The Federal government would be similarly helpless, perhaps even more so, as their frontier responsibilities were far greater than those of the CSA.