Rev 6:9 Καὶ ὅτε ἤνοιξεν τὴν πέμπτην σφραγῖδα, εἶδον ὑποκάτω τοῦ θυσιαστηρίου τὰς ψυχὰς τῶν ἐσφαγμένων διὰ τὸν λόγον τοῦ θεοῦ καὶ διὰ τὴν μαρτυρίαν ἣν εἶχον. (NA28)
Rev 12:11 καὶ ⸀αὐτοὶ ἐνίκησαν αὐτὸν διὰ τὸ αἷμα τοῦ ἀρνίου καὶ διὰ τὸν λόγον τῆς μαρτυρίας αὐτῶν καὶ οὐκ ἠγάπησαν τὴν ψυχὴν αὐτῶν ἄχρι θανάτου. Revelation 12:11 (NA28)
Rev 13:14 καὶ πλανᾷ τοὺς κατοικοῦντας ἐπὶ τῆς γῆς διὰ τὰ σημεῖα ἃ ἐδόθη αὐτῷ ποιῆσαι ἐνώπιον τοῦ θηρίου, λέγων τοῖς κατοικοῦσιν ἐπὶ τῆς γῆς ποιῆσαι εἰκόνα τῷ θηρίῳ, ὃς ἔχει τὴν πληγὴν τῆς μαχαίρης καὶ ἔζησεν Revelation(NA28)
The argument in the book is a touch complex so I can't really explain why he argues for this beyond that he says the concept of dying in Revelation as a metaphor for conversion. Those killed by God and his agents are those who come to be persuaded by the gospel; those killed by Satan are those who turn to him.
Theology aside, it raises the question of whether δια + acc is reasonable as an indicator of instrumental cause.
BDAG is cited
This ties in with BDFα. w. acc. of thing (schol. on Pind., N. 4, 79a; 2 Macc 12:11; EpArist 77) διὰ τὸ αἷμα by the blood Rv 12:11. διὰ τὰ σημεῖα by the miracles 13:14.
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 226). Chicago: University of Chicago Press.
Given the limited range of examples given for such a nuance I am wondering whether folk can see this in their reading of classical / other koine?222. Διά with accusative. In a local sense ‘through’ (classical only in poetry, Hellenistic in prose since Dionys. Hal. [Käser 54]) only Lk 17:11 διήρχετο διὰ μέσον (SBL, A al. διὰ μέσου, D μέσον without διὰ [§215(3)]) Σαμαρείας καὶ Γαλιλαίας, hardly correct. J. Blinzler, Festschr. A. Wikenhauser (Munich, 1954) 50ff. holds that μέσον Σαμαρείας καί is to be struck out as a gloss; on the impossible geography s. H. Conzelmann, Die Mitte der Zeit 60–6 [68–73]. Otherwise ‘because of, for the sake of’, for the reason (propter) as well as for the purpose (= classical ἕνεκα; cf. §216(1)), so that the MGr meaning ‘for’ also arises (Hatzid. 212f.): Mk 2:27 τὸ σάββατον διὰ τὸν ἄνθρωπον ἐγένετο καὶ οὐχ ὁ ἄνθρ. διὰ τὸ σάββατον, Jn 11:42, 12:30, 1 C 11:9 etc. (already in some books of the LXX the only use of διά with acc. [Johannessohn II 240f.]). ‘By someone’s merit’ (as in classical) Jn 6:57, R 8:20, therefore also ‘by force of’ Rev 12:11, 13:14.—Mayser II 2, 368f., 426; Humbert 144ff.; Ljungvik, Syntax 33ff.; Ghedini, Vang. ap. 452; Rob. 583f.; Moule 54f.
Blass, F., Debrunner, A., & Funk, R. W. (1961). A Greek grammar of the New Testament and other early Christian literature (p. 119). Chicago: University of Chicago Press.