This is a tough one. I have to think about this.
Finding a place to eat breakfast on Sunday morning anywhere in the South is a real challenge. Until 11:00 a.m. rolls around, I almost always have to subsist on chain restaurant food, such as Cracker Barrel, etc.
That said, although I haven’t been there in fifteen years, in Bowling Green, Kentucky you might try Murray’s a few hundred feet away from the intersection of the U.S. 31-bypass that goes around town & U.S. 231. I drove by there in June, and I noticed that another name was on the building, but the sign also said "formerly Murray’s."
Unless things have changed, I don’t believe Judy’s Castle (almost across the street) is open on Sundays. But it couldn’t hurt to check.
In Cave City, Kentucky there is a small restaurant chain called Jerry’s. They have pretty good food. At least that is my experience with any Jerry’s restaurant I have ever dined in.