I've assumed from long-casual reading that it meant "a conflict over something relatively trivial." But today's ubiquity prompted me to go digging a bit further.
The OED fails to mention this meaning, which briefly made me wonder if I had it all wrong.
bun-fight n. a jocular expression for a tea-party (cf. tea-fight n. at tea n. Compounds 3).
1928 R. Campbell Wayzgoose 7 It [the wayzgoose] combines the functions of a bun-fight, an Eisteddfod and an Olympic contest.
But it was baffling to think my friends were calling Clearing an expression of civility.
Collins does better with meaning #2 being "a petty squabble or argument".
Following up on a tea-fight via the OED....
tea-fight n. colloq. or slang humorous name for a tea-party or tea-meeting.
1849 A. R. Smith Pottleton Legacy xxxv, Their various small parties—‘tea-fights’ as young Grant called them.
1901 Scotsman 5 Mar. 7/5 The good people..organise a splendid weekly tea-fight and concert for our behoof.
bun-struggle n. = bun-fight n.
1899 R. Whiteing No. 5 John St. vi. 53 She wants yer to show up at a sort o' bun-struggle in 'er room..kind of a tea-fight.
bun-worry n. = bun-fight n.
1889 A. Barrère & C. G. Leland Dict. Slang Bun-struggle or worry (army), a tea meeting; an entertainment [for] soldiers in a garrison.
1911 W. De Morgan Likely Story 224 Madeline..had been going to a Bun-worry.
And as a bonus
† tea-shine n. colloq. Obs. a tea-party (cf. tea-fight n.).
1838 J. W. Carlyle Lett. (1883) I. 98 Two tea-shines went off with éclat.
A bun-fight Ngram: the rise of "bunfight", although without distinguishing between its senses.
Another person to briefly look at the subject observed the nineteenth-century terms "crumpet-scamble" and "muffin-worry" as synonyms for "bunfight", in the sense of "tea party".
It's not clear than anyone has bothered digging back to exactly where the argument meaning was first documented, but presumably it was post-'20s.