As it happened, no, it couldn't... but suppose in early 1862, the Confederacy had abandoned its attempts to defend all the states and the Mississippi, and concentrated to defend the "core" area. Would that core area then have the wherewithal to sustain itself until (say) 1866?
I guess it depends on when in 1862. I'll propose after Shiloh. While the fall of Donelson had broken the defensive position, the failure at Shiloh seemed to guarrantee that the rail hub of Corinth would fall. Van Dorn strips Arkansas/Missouri of troops just as before. Curtis is on the prowl after winning at Pea Ridge. Memphis is lost. New Orleans and Baton Rouge are in Federal hands.
The problem in defending a core and in allowing the other places to fall is that the regions given up at the outset are now a springboard for the next Federal move. That moves up the timetable for operations against various Confederate ports along the Gulf Coast, particularly Mobile. The Union could push control deeper into Arkansas much sooner as well, relieving some of the pressure of disruptive raids into Missouri from northern Arkansas.
The positive for the CSA is that it now has larger mobile forces with which to resist Union initiatives for 1863 and that gives it advantages if it can concentrate against one or another Union initiative. But this also means that the Union timetable has moved up by the better part of a year and the Union has more forces available for offensives as well (not likely proportionately larger, which would allow shifts of CSA forces to obtain local advantages.) The problem for the CSA is that it's interior is more threatened, and its infrastructure no better so that it would be hard taxed to move the extra men for dynamic defense. That might not be critical though. Perhaps a Federal army would be routed somewhere in the West in late 1862.
Jackson, MS, and Little Rock, AR would likely fall before the end of 1862. Jackson is another rail hub, so that will hurt. I would expect Mobile Bay's forts to fall in 1863. This would put increased pressure on Wilmington and Charleston.
I suspect the impact of morale in the T-M would be awful, damaging the ability to field an effective army in the theater. It isn't clear that it would necessarily reduce guerrilla operations in Missouri, but cavalry raids would be harder to conduct with the CS forces backed into Southern Arkansas, so the guerrillas would have less direct CS support. The Missouri State Militia cavalry formed in early 1862 might handle the load in Summer of 1862 and not require the call up of statewide EMM militia that fed CSA recruiting in the state. In Arkansas and Texas the Camden and Red River expeditions respectively would be possible a year earlier (not that those went well for the Union...)
So it really depends on what the CSA can accomplish in the West in 1863 with a more dynamic defense/open position and more heavy pieces on the board as we like to say in chess. To compensate for the other losses it would need to hold Chattanooga throughout and probably rout/capture the better part of a large Federal army in the theater, perhaps two armies. That's a tall order.