Yet another source, now that I search.
OLYMPIODORUS, In Platonis Alcibiadem commentarii
Τί οὖν τὸν αἰσθανόμενον χρὴ ποιεῖν; > ἐρωτᾷ αὐτὸν τί δεῖ ποιεῖν πρὸς τὸ μαθεῖν· ὁ δέ φησιν οὐδὲν χαλεπὸν οὔτε τὰ τοῦ Ὤτου καὶ Ἐφιάλτου ποιῆσαι (‘Ὄσσαν ἐπ’ Οὐλύμπῳ μέμασαν θέμεν’), ἀλλὰ δοῦναι λόγον καὶ λαβεῖν. καὶ Ἀρχιμήδης μὲν μετὰ τὴν εὕρεσιν τοῦ βαρυούλκου, ἐν ᾧ ἔδειξεν ὅτι τῇ τυχούσῃ δυνάμει τὸ τυχὸν βάρος κινήσει, ἐπειδὴ ὅσον ἐπὶ τούτῳ καὶ εἷς ἄνθρωπος ἠδύνατο κινῆσαι τὴν γῆν, μέγα ἐφθέγξατο καὶ ἀδύνατον, ὅτι ‘πᾷ βῶ καὶ κινῶ τὰν γᾶν’· ἀδύνατον γὰρ μὴ εἶναι ἐν τόπῳ· Σωκράτης δὲ εὐτελῆ αἰτεῖ.
So here it’s a βαρυουλκός/βαρουλκός, which LSJ gives as a “lifting-screw.” The question-version of the story seems to make more sense here than anywhere else. So to me this seems more interesting than the Simplicius version.
Beyond that, a contemporary of Tzetzes also seems to have known the story with a χαριστίων. Michael Italikos, probably the same source (Simplicius ?) as Tzetzes.
Ἀρχιμήδης δὲ λῆρος διὰ τὸν χαριστίωνα, τὸ πολύσπαστον ὄργανον, ὑπερόγκους λόγους φθεγγόμενος· πᾷ βῶ καὶ κινῶ τὰν γᾶν.