The verb is found in the proverb collector Zenobius (4.50), whom the Suda cites verbatim. It is frequently commented on by lexicographers, grammarians and other paroemiographers, e.g.: Diogenianus, Hesychius, the Lexica Segueriana, Photius, Eustathius, Gregorius (Patriarch Gregory II), Michael Apostolius. The Suda's infinitive, here, was once regarded as an unattributable fragment of Attic comedy: adespota fr. 1036 Kock, but not registered by K.-A.
The verb clearly derives from ke/rkwy (i.e. only indirectly from ke/rkos): either the mythical man-monkeys (kappa 1405, kappa 1406), or the more prosaic long-tailed ape of the same name. (The ambiguity of the noun is noted in kappa 1410.) LSJ s.v. derives the verb from the latter meaning: "play the ape", not "the Kerkops".
The lexicographic tradition of the verb diverges around two forms: the infinitive attested here, and the plural present participle kerkwpi/zontes given in kappa 1409 (q.v.). The major tradition from Zenobius onwards is the infinitive, which is glossed around notions of malice (be treacherous, deceive, flatter), consistent with the description of the Kerkopes, and the first gloss of ke/rkwy in kappa 1410, as "cheat".
The participle kerkwpi/zontes is given in Hesychius and Photius (cf. kappa 1409), and is associated with mockery. The second sense is consistent with the perception of monkeys and apes as mimics, and the association is made explicit from Photius onwards.
It seems likely, given this divergence, that different lexicographers were merely guessing at the meaning of the verb from the time of Zenobius (whose etymology from ke/rkos was already fanciful). One tradition guessed that it involved the Kerkopes, hence deceit (or flattery, through Zenobius' wagging tail); another guessed it involved the kerkops ape, hence mimicry. Because the sense "ape" appears to be late (see kappa 1410), Hesychius' derivation is probably wrong. Semonides' kerkwpi/a (fr.34) is glossed in LSJ as "trickiness", which is consistent with ke/rkwy = "knave", the sense used here.