Here's an edifying story re "the Classical Guitar that my wife found at Goodwill for $15 is developing a split in the back--"

When my beloved guitar suffered an accident which caused a split in the top, I raced to the phone and called my brother, the expert. He said, "This is the important part. Does the crack cross a glued joint?" The answer was no.

Acting upon his advice, my husband and I put the smallest bit we could use in the drill and drilled round holes at the ends of the splits. (He operated the drill, I held the guitar steady.) The holes needed to be just slightly bigger than the width of the splits. Since my splits were narrow, our "bit" was a sewing-machine needle.

Why? A round hole at the end of a split stops it from propagating.

Then I went to the hardware store and bought some wood putty to match the guitar. I filled up the round holes and pushed it into the splits as best I could. (The putty wasn't necessary to the sound of the guitar, I just thought it looked better.) The guitar has played as well as ever since then.

Perhaps you can apply this cure to the classical guitar.