“I don’t care if it harelips the governor” (or “hare-lips the governor”) means the same as “come hell or high water.” The phrase (also popularly used in to “harelip hell”) is of undetermined origin, but has been popular throughout the South in the 20th century.
       
       
Google Books
Collier’s: Incorporating Features of the American Magazine 
1946
Pg. 257:
“No. I’ll find a place, if it harelips the governor.
     
2 March 1969, San Antonio (TX) Light, “The Don Says: ‘Red’ Keeps Plugging Away,” pg. 12-C, col. 2:
The “Old South” will never rise again, but the Free State of South Texas just might.
 
Old Red Berry, the retired gambling man and active sportsman, plans to have horse-racing in Texas one way or another even if it hare-lips the governor.
   
Google Books 
The Hellions
By George McNeill
New York, NY: Bantam Books
1978
Pg. 15:
“Yessir, Slade, you and Fred and Courtney and the rest, you better than any [n-word—ed.] ever lived, and when you do your work with that Scaggin [n-word—ed.] you goin’ to have a swell supper as a reward, no matter the cost, don’t care if it harelips the governor.”
 
31 August 1983, San Francisco (CA) Examiner, “Bears must face A&M’s revival of the 12th man,” pg. F4, col. 1:
Thus, if Cal wins the toss next Saturday, the Bears will line up opposite a kicker and 10 non-scholarship athletes whose grimaces would harelip the governor (More Lone Star talk. Translation: Looking fiercely determined).
     
Google Books
Texas Crude:
The how-to on talkin’ Texan
By Ken Weaver
New York, NY: E. P. Dutton, Inc.
1984
Pg. 2:
“If it harelips the governor . . .”
No matter what the cost. Equals “Come hell or high water” and implies an implacable determination to succeed in any endeavor, from working a crossword puzzle to finagling the purchase of a select oil lease, even if to do so constitutes a slap in the face of the law. “I know she’s married, and I know she loves her husband, and I know he’s a big, mean, jealous man, but i’m gonna bed her if it harelips the governor!”
 
Mavens’ Word of the Day
June 25, 1998
harelip, v.
(...)
The verb harelip (as it’s usually spelled) is found chiefly in figurative phrases that suggest dire consequences that one will ignore in order to get something done. If it harelips (something) as a phrase thus means ‘regardless of consequences’ or ‘come hell or high water’. The meaning of harelip alone can’t really be defined, but ‘to destroy’ provides a rough guideline.
 
One source, referring to Alabama in the 1920s, defines to hare-lip hell as “to show great determination regardless of consequences. ‘I’m gonna do it if it hare-lips hell.’” From Texas we have an example of “I’m gonna bed her if it harelips the governor,” and from the Ozarks, “I’m going to do it even if it harelips all the hogs in Texas.”
 
This expression is first recorded in a 1960 collection of regional expressions, although various sources claim that it was in use in the 1920s, the 1940s, and the 1950s. It seems to be confined to the South. It is presumably based on harelip, a sixteenth-century term for ‘a cleft lip’ (a congenital defect of the upper lip in which a fissure extends into one or both nostrils); this term is now sometimes considered offensive.
 
Google Books
Ride the Devil’s Tail
by Dan Parkinson
New York: Kensington Publishing Corporation
1990
Pg. 174:
”...back if it hare-lips half of Texas.”
 
This Eclectic Life
Thursday, March 29th 2007
Thursday Thirteen #3: A Lesson in Texas Language
(...)
If someone says, “I don’t care if it hare-lips the governor,” that usually means that they don’t care about the outcome at all! You don’t have to use the whole phase; it is just as effective to say, “I mona eat that third piece of cake if it hare-lips the governor.” Around here, we will know what you mean.
 
A Way with Words
Harelips the Queen
Posted by grantbarrett on May 9, 2014
“I don’t care if it harelips the queen” means “come hell or high water,” or “regardless of the consequences.” The phrase is particularly popular in Texas, as are such variants as harelips the governor, harelips the president, harelips every cow in Texas, harelips the Pope, harelips the nation, and harelips all the cats in Grimes County, among many others. Harelip refers to the congenital deformity known as a cleft palate, which resembles the mouth of a rabbit, and is sometimes considered offensive.