It sounds more like a fungal disease that attacked part of the root system.

Texas Root Rot will do a partial kill like that.

http://www.artistic-arborist.com/sept_2003.htm
http://www.ehow.com/list_5790814_pepper-tree-diseases.html

Replace it with a resistant species.

Like lazygardens said, this tree is seriously weak to TRR, and there's no treatment.

This year is going to perhaps be the hottest and the driest yet. We're in trouble and so are "legacy" imported plants that could grow way back when there was water in the rivers.

From much experience in killing plants I'd say you could nurse it for years but the peeling bark thing says it's really done for as a landscape tree. The link below pretty much covers it.

Here is a link that might be useful: U of A Cotton Root Rot notes

Ugh, well thank you so much for your answers. Trees are like people to me. I know it sounds stupid, but I really get attached to them. Also I absolutely hate seeing someone chop down a healthy, lovely tree. :-( I won't replant another tree in that spot. Unfortunately this tree provided a lot of shade for the south side of our house.

Consider an arbor for the south side, with a deciduous vine, or a rot-resistant tree.