Lee went north in 1863 in no small part to prevent his army from being split and part sent west in a futile effort to save Vicksburg.
I agree with you. Splitting up the ANV in the hopes of creating a pickup Army to what? Go at the Army of the Cumberland. George Thomas was running a corps in that Army and would eventually command it. He was a tough nut. As Chickamauga, Missionary Ridge, Nashville & Franklin would later prove.
It might have worked, but at the time it would have looked like a long-shot.