The honest answer? We have no idea, but probably not. That said, he is an authentic name from the era.

Achilles is found, in the form A-ki-re-u.

The reason for this rather odd spelling is that Linear B is a writing system that was not originally created for Mycenaean Greek, but likely for the Minoan language. Thus, symbols, letters, and sounds have been forcibly shoe-horned by scribes into the Mycenaean.

In one of its attestations in Linear B, it is written in the dative A-ki-re-we, and is a name is attested to on some of one of the Pylos tablets (FN79.2 specifically). Sourced from page 198, the Appendix of Individuals and Society in Mycenaean Pylos (Brill, Boston, 2013), by Dimitri Nakassis. Also, reference page 104 of Documents in Mycenaean Greek by Michael Ventris and John Chadwick (Second Edition. Cambridge, 1973), which attests to the name at Knossos (KN Vc 106) as well.

If I'm not mistaken (I can't find my reference for this at this moment) sources attest to "A-ki-re-u" in the Pylos tablets (I can't find it in Nakassis at the moment, but I know I saw it...somewhere) as being a shepherd of some local importance, his importance attested to due to his mention in palatial documents, not the great warrior you're expecting. Never mind that he was geographically in the wrong place, considering that the "historical" Achilles - that is the shepherd - came from Pylos, and the Iliad's Achilles was a Thessalian, of a distinct geographic area.

Someone far more versed in the Late Bronze Age, Homer, and the Ancient Greek language can probably add a bit more detail, but I believe this is basically the extent of the archaeological record of what we've found for that name.