Joe Wright, who had played a key role in setting the young Tolkien on the philological path that would occupy him to the end of his days, produced a stunning six-volume work of scholarship, The English Dialect Dictionary, from 1898 to 1905, in which Volume I contains a substantial entry for attercop (91).
Some years later, Tolkien might have seen the poem by Robert Graves, "Attercop: The All-Wise Spider," published in 1924.
Returning to real etymology, the first element in attercop goes back to Old English ator (variously spelled, ater, attor, etor, etc.), meaning "poison"; cognate forms Tolkien would have known from the other medieval Germanic languages include Old Norse eitr, Old High German eitar, Old Saxon ettar, hettar.
We have thus found a kind of double meaning--"head" versus "cup"--is the second element of attercop. Spiders are both "heads of poison" and "cups of poison," and they may even have "bags of poison" or deliver a "pox of poison." And this is not the end of our odyssey through Tolkien's wonderful words and their double-or even triple-meanings.
(9) As it happens, there is a Finnish word, myrkky, which is quite close phonologically, but which doesn't mean "dark" at all; rather, it means "poison," just like the first element of attercop. Cognate to these are Estonian murk, Saami (Lappish) mir'hku, and Hungarian mereg, all meaning "poison"--and all looking like the first element in Mirkwood.
(3) Where Tolkien writes, "Attercop! / Down you drop!", it's not too far-fetched to think Tolkien could be punning on the Welsh, pointing out (beyond the literal meaning of the lines) that Old English ator-coppe eroded ("dropped down") into Welsh adrop.