I mentioned a few years ago (in a comment beneath the posted question) that there is one relatively old instance of flitter-lip in a brief news item in the [Winnsboro, South Carolina] Fairfield News and Herald (August 8, 1888):
TRIAL JUSTICE COURT.—Before Trial Justice Cathcart on Friday the case of Louisa Pagan, assault and battery on Jim Hamilton, better known as 'Flitter-lip Jim' was heard. Verdict not guilty.
Although there is no explanation in the item of how Hamilton acquired his nickname, Dictionary of American Regional English, volume II, D–H (1991) notes that flitter sometimes appears as a variant form of flutter, suggesting that flitter-lip might carry a meaning along the lines of "loose-lips" or "flapjaw"—approximately, "someone who talks to much."
More to the point of the posted question, however, is another DARE entry for flitter:
flitter intj Used as a mild expression of annoyance [Cited instances 1954 Harder Coll. c[entral] w[estern] TN, Aw flitter. 1968 DARE (Qu. NN8a, Exclamations of annoyance or disgust: "Oh ___________, I've lost my glasses again.") Inf[ormant] LA28, Flitter.
The two cited instances of this interjection are from central western Tennessee and from Louisiana—two noncontiguous (but not terribly distant) states in the western part of the Old South. Oklahoma (where the poster's father lived at one time) is farther west but again not very distant from either Tennessee or Louisiana. One possibility, then, is that "flitter lip!" used as an interjection is simply an extended form of "flitter!" used in the same way.
Another possibility is that flitter-lip may be a variant of flutter-lip, which may convey any of several specific slang meanings, although I have not found these recorded in slang dictionaries. One use of the term, from Willie Nelson, The Facts of Life: and Other Dirty Jokes (2002) implies stammering or stuttering:
I came to town [Nashville, Tennessee] with "Night Life," "Mr. Record Man," "Crazy," "Funny How Time Slips Away," and a few others. Hank [Cochran] had heard some of my songs one night at Tootsie's Orchid Lounge in Nashville. Everyone hung out at Tootsie's. She loved all the crazy people, Fluffo and Flutter Lips (better known as Wayne Walker and Mel Tillis), Faron Young, Hank, Charlie Dick (Patsy Cline's husband), Little Jimmy Dickens, and Billy Walker.
Nelson doesn't explain the nickname "Flutter Lips," but Mel Tillis was famous for having a serious stutter except when he sang (which he did very well).
Possibly to the same effect is this instance from Ramon Adams, The Old-Time Cowhand (1948/1961):
When the [auguring] match came off, our champion, havin' some talent for poetry, started off with such giggle talk as, "The hoss he neigh, can you tell what he say? The cow she moo, the bull does too' the dog he bark, till the moon goes dark; the lion he roar, till his throat gets sore; the catamount squall like a bronc in the stall; the rabbit he run, 'cause he's a-skeered of a gun; the kyote yip, like he's got the flutter-lip." He goes on like this till he runs plumb through the animal kingdom.
The "old-time cowhand" narrating this story is, I believe, from Texas. In context, the narrator seems to be equating the coyote's "yip-yip-yip-yip-yip-yip" call with a stutter.
Another uses the term to describe certain vocalizations or musical sounds that are not actual words. From Heidi Von Gunden, The Music of Pauline Oliveros (1983) [combined snippets]:
The third electroniclike motive is a series of percussive envelopes titled "lip pops," "tongue clicks," "snap fingers" and "flutter lips." Although these sounds resemble childlike sonic play, they are used in a serious context and are difficult to perform because they demand strict ensemble precision.
...
The second phrase introduces muting while mixing ring modulation with white noise (as in "shung" sung by the tenors in measure 6), and the third phrase features percussive sound and new attacks. "Wht" becomes a version of white noise; "bbbbbbbbt" (flutter lips) is a new percussive sound, and " pow-wowōwowōwowō" is another ring modulation. The separate dynamic markings for each sound and frequent tempo fluctuations clearly define the phrase structure so that the listener recognizes an initiation and conclusion of musical ideas.
Similarly, Cleo Coyle, Bulletproof Barista (2023) uses "flutter-lipped" to describe to the "pththththt" sound also known as a Bronx cheer:
"I'm high on life," Jerry cracked, adding a flutter-lipped Bronx cheer. So give us a kiss then."
I don't know which of these possibilities (random expansion of flitter used as an interjection, variant of flutter-lip in the sense of "stammerer," or variant of flutter-lip in the sense of "Bronx cheer-like sound") is the most likely source of flitter-lip as used by the poster's father, but I suspect that one or more of these sources may have influenced his usage.
