The claim that ABS and LLR stand for "Associate Back Scratcher" (not Associated as in the source you found) and "Licensed Log Roller" seems to be quite widespread. It appears in, among others, Faye Hammill's Women, Celebrity, and Literary Culture between the Wars; Bill Peschel's Writers Gone Wild: The Feuds, Frolics, and Follies of Literature's Great Adventurers, Drunkards, Lo vers, Iconoclasts, and Misanthropes; and assorted blog posts. Unfortunately, none of these books or internet pages cites a source for this claim, although they do provide slight circumstantial evidence by mentioning it as part of an argument that Gibbons was thumbing her nose at the literary elite and satirising established practices in literature.
The most authoritative source I've been able to find is Lynne Truss's introduction to the 2006 edition:
Gibbons also overtly rejected the literary world with her dedication to Pookworthy - A.B.S. and L.L.R. stand for 'Associate Back Scratcher' and 'Licensed Log Roller'. She didn't move in literary circles, or even visit literary squares, or love in literary triangles.
Since this appears within the book itself, some might take it to be conclusive. But of course Truss is only one reviewer, and her word isn't necessarily worth more than anyone else's just because she managed to get her review included in a later edition of the book.
I suspect the real answer is that
we don't know. The very fact that you've found different sources making different claims about what the initials stand for suggests that there is no 'canonical' answer. Apparently Gibbons never said explicitly what "ABS, LLR" stood for, and others have taken the interpretation of these initials into their own hands. It's even possible that she had no particular words in mind to fit these initials - that, essentially,
there is no answer. If she did, surely she would have mentioned them either in the book itself or in some kind of supplementary materials? The initials already convey the idea of "fancy pointless acronyms" even without knowing what they stand for, and perhaps that's all they were intended to do.