If the etymology of λαίλασ ("ὁ μὴ ἐκ γένους τύραννος, Cyr., Suid.; Lydian acc. to Hsch.", LSJ) is correct to compare λαός, which may be debatable, the title could be read accordingly, which should be fairly convincing, I think, in face of the rhyme with βᾰσῐλεύς, except that's uncertain as well. This would leave two questions that I won't be able to answer, a) namely the nationality of Achilles or at least the provenience of the myth, and b) the meaning of the first argument.
In addition, it is notable that wanax as an earlier synonym is hotly debated. In general this is a good idea, follow the money, except that different ideas of rule have to be considered. So while tyrant may have continued where λαίλασ broke of, this would suggest a warrior king. So, as unlikely as it might be, it's an important problem space to consider.
As for
a) if it belongs to the same stratum as basileus, it may indicates Egyptian. Basileus is like behemoth speculatively maybe prefixed with an Afroasiatic determiner, which is particularly pertinent with plural of honor. There is no historic framework to date it that I'm aware of, but there are records of placenames to show this at some later stage. Within the Trojan War setting.
b) Achaean sounds to me like the most likely match for the first argument. The meaning is clear enough, though the etymology is not. This is compatible with ἄχος to the extent that for example res public, blodthing and Reichstag and so on have bleached semantics, so criticism such as Beekes' towards a finer distinction requires to show that "distress" or "pain" is a secondary development, ie. "matter" in "what's the matter with you".