This root first appeared as unglossed ᴹ√MŌ in The Etymologies of the 1930s with derivatives like ᴹQ. mól/N. mûl “slave, thrall” and ᴹQ. móta-/N. muda- “labour, toil” (Ety/MŌ). The root √mō was mentioned again in notes associated with 1950s Quenya prayers, where it was glossed “labour, be afflicted” along with verb Q. moia- of similar meaning and noun (Q. or primitive) mōl “slave” < mō-l; Tolkien went on to suggest √mol might be a better root, though he marked it with a “?” (VT43/31).
In notes on “large vs. small” from 1968 Tolkien had (root or verb?) mol- “labour” and (noun?) mōle, followed by MBOL/BOL with derivative Q. molda “big, large”, but this note was crossed through and √(M)BOL “✱large” did not appear in the next list of the large/small roots (PE17/115). Thus I think √MO(L) reverted back to the meaning “labour”.
This root for licking had precursors dating all the way back to the Qenya Lexicon of the 1910s, where it appeared as ᴱ√LAVA “lick” with derivatives like ᴱQ. lava- and G. lav- of the same meaning (QL/52; GL/53). In The Etymologies of the 1930s it became ᴹ√LAB “lick” (Ety/LAB), and √LAB continued to appear in Tolkien’s later writings with the glosses “lick” (PE17/72; PE22/151; PE19/92) or “lick, move the tongue” (VT39/15; WJ/416). It was thus quite stable in Tolkien’s mind.