Etymonline.com notes
"Shat is a humorous past tense form, not etymological, first recorded 18c."

So it's pretty old, but was made up to be funny.
( Moon Girl Technologies horrendous beanbag, Mon 1 Jun 2009, 13:54, archived)

This.
I know it's not the correct past tense. My dad says it, and I used to find it hilarious when I was little.
( Esme Weatherwax I swear on my tash it's daycent hash, Mon 1 Jun 2009, 13:57, archived)