Note the Hebrew words that τελέω translates in the LXX:

In Psalms 115:28 (116 in English translations) ἐτελέσθησαν in LXX translates וַ֭יִּצָּ֣מְדוּ
niphal, imperfect waw consecutive, 3rd person, masculine, plural of צמד
This is the second most common Hebrew lemma τελέω translates. Note the meaning of צמד
given in the Logos Bible Software (LBS) chart.
Here's an LBS chart of the words LXX uses to translate צמד.

This is the closet I found to the Hebrew meaning in BDAG:
Diagoras of Melos in Sext. Emp., Adv. Math. 9, 55 κατὰ δαίμονα κ. τύχην πάντα τελεῖται=‘evervthing is accomplished acc. to the will of the god and of fate’; an anonymous writer of mimes [II AD] in OCrusius, Herondas5 [p. 110-16] l. 175 τοῦτο τετέλεσται); Cf. τετέλεσται used absolutely in vs. 30 (cf these two verses are to be taken as referring to the carrying out [s. 2 below] of divine ordinances contained in the Scriptures, cf. Diod. S. 20, 26, 2 τετελέσθαι τὸν χρησμόν=the oracle had been fulfilled;
Arndt, W., Gingrich, F. W., Danker, F. W., & Bauer, W. (1979). A Greek-English lexicon of the New Testament and other early Christian literature : a translation and adaption of the fourth revised and augmented edition of Walter Bauer’s Griechisch-deutsches Worterbuch zu den Schrift en des Neuen Testaments und der ubrigen urchristlichen Literatur (pp. 810–811). Chicago: University of Chicago Press.
The TDNT has a paragraph of the LXX's use of τελέω to translate צמר:
- The religious sense is presupposed, but only with ref. to heathen cults and negatively, in the mid. pass. “to dedicate oneself to the service of a god” ἐτελέσθη Ισραηλ τῷ Βεελφεγωρ Nu. 25:3, cf. ψ 105:28, of individuals τετελεσμένον τῷ … Nu. 25:5 (all 3 passages for צמר ni), τετελεσμένη the “qedesha” Hos. 4:14; for the LXX קדשׁ is connected with τελτελεσφόρος and τελισκόμενος “consecrated,” Dt. 23:18; also τελετή 3 Βασ. 15:12
Delling, G. (1964–). τέλος, τελέω, ἐπιτελέω, συντελέω, συντέλεια, παντελής, τέλειος, τελειότης, τελειόω, τελείωσις, τελειωτής. G. Kittel, G. W. Bromiley, & G. Friedrich (Eds.), Theological dictionary of the New Testament (electronic ed., Vol. 8, pp. 58–59). Grand Rapids, MI: Eerdmans.
Thus, “to dedicate oneself to the service of a god [Baal of Peor]” appears to be the best meaning of ἐτελέσθησαν τῷ Βεελφεγωρ.