People are being a bit too literal, methinks. I half-expect people to assume by the use of the phrase "Fabian Strategy" that I'm insinuating that the Confederacy should have fought with spears and shields rather than Enfield muskets. I'm simply referring to a strategy of general avoidance of major battles (especially offensive ones) in an effort to avoid high casualties and seek to wear down the Union will to fight. In other words, the opposite of Robert E. Lee's general strategy, which was to destroy the main Union field army in a decisive battle.
There are several problems with a Fabian strategy.
First, it is often politically unacceptable and demoralizing. It is usually only applied after repeated military disasters when people realize that there is no alternative. For the Confederacy to apply it off the bat might collapse the entire war effort.
Second, it concedes significant sections of the Confederacy to the Union immediately. Although many people don't want to admit it, there is significant support for the Union throughout the Confederacy, particularly in the Appalachia country. Without significant Confederate resistance, the Union is likely to occupy all of Tennessee, Arkanasas, northern Alabama and Mississippi and much of Louisiana far quicker than it did IOTL. Having access to eastern Tennessee, northern Alabama, and some of Arkansas so quickly allows the Union to recruit from those areas far quicker.
Third, at some point, you have to fight. The Confederacy mainly fought a defensive war anyway. So what battles are they exactly avoiding? What do they give up and where do they choose to fight? It's hard to see how exactly the Fabian strategy is to be implemented.
Fourth, the purpose of the Fabian startegy is to preserve your strength until such time help arrives either by a foreign power or by building your own army. It worked for Rome because Hannibal could not get significant reinforcements. It worked for Washington because the British could only send so many troops, and eventually French entry into the war prevented that. The Confederacy does not have those advantages. The Union is right next door. The Union can keep sending in more and more armies. Whatever they seize, they'll likely keep. Unlike the Carthaginians or Redcoats, the Union has significant support. Factoring in pro-Union whites and slaves, something like 50% of the total population is for them. They can recruit plenty of local help. And even among the loyal Confederate population, there will always be a significant collaborationist sentiment once occupation happens. Furthermore, when the Union advances, they deny the Confederates the recruiting pool for their own armies. The hotbeds of Confederate fire eaters are those states with the lowest white population and lowest industry. Once Europe sees the Confederates surrender huge swatches of territory without heavy opposition, they are even less likely to intervene than they were previously. Instead of buying time to strengthen their forces, the Confederates actually do worse.
Fifth, the guerilla has been vastly overrated. Guerilla forces, alone, are almost always defeated. Guerillas only work when combined with a real army that fights conventional battles.
Sixth, avoiding the battles that gave Robert E Lee high casualties also means the the Union avoids high casualties. Seeing Confederate armies retreat, earlier liberation of Union strongholds, and light Union casualties erodes much of the anti-Lincoln and anti-war sentiment in the North.
I don't see the elements of a successful Fabian strategy being present. There are certainly battles the Confederates would be better off avoiding, but that can only be known with hindsight.