The late David Flusser relates the Pauline discussion on EIDWLOQUTON to the brief passage in Did. 6:3, which advises neophytes to "be on your guard against food sacrified to idols (EIDWLOQUTOU), for it is the worship of dead gods (LATREIA GAR ESTI QEWN NEKRWN)." Not only does this verse begin with similar wording as 1 Cor. 8:4-6 ("as to the eating of food sacrificed to idols"), but it shares the same perspective that "gods" do not exist.
In my opinion, Paul probably draws on Deutero-Isaiah's satire on idolatry (Isa. 44), and states that "there is no idol in the world and there is no God but one" (= Isa. 44:8, "Is there any other God besides me? [[There is no Rock, I know of none]]"); the use of EIDWLON to refer false QEOI or LEGOMENOI QEOI has a precedent in the use of QEOS to refer to idols in Isa. 44: "Whoever fashions a god (PLASSONTES QEON)" (v. 10), "and with the remainder he makes gods (EIRGASANTO QEOUS)" (v. 15), and especially "with the rest he makes his carved god (EPOIHSAN EIS QEON GLUPTON)" (v. 17), so if Paul was influenced by Isa. 44, he would have found an equation between "idols" and so-called "gods".
Flusser argues that Did. 6:3 makes the same argument that "idols are nothing" and other gods do not exist by referring to the intended recipients of sacrifices as "dead gods" (dead gods = gods that are not in existence). In this case, the OT allusion is to Num. 25:1-3 and Ps. 106:28 ("they attached themselves to Baal of Peor and ate sacrifices of the dead"), which in subsequent interpretation became "sacrifices to the dead" (cf. Jub. 22:17, m. 'Abodah Zarah 2:3), and OT references to idols as inanimate and lacking senses (cf. Deut. 4:28, Jer. 10:3-5).
So one possible solution is to understand that (in light of Isa. 44) Paul uses EIDWLON as a metaphor for LEGOMENOI QEOI, since a "carved god" (which the carver calls "my god", Isa. 44:17) is an "idol". Thus, Paul could say that idols have no existence. Similarly, gods have no real existence. This "knowledge" (1 Cor. 8:7) then becomes the underlying premise for Paul's counsel on not "stumbling" other brothers in v. 7-13, for not all brothers share the same "knowledge" (i.e. understand that "so-called gods" do not exist). Then, after the digression in ch. 9, Paul returns to the subject of sacrifices because some in the church have fallen into idolatry (1 Cor. 10:7), and like the Israelites in the wilderness this has brought divine displeasure (compare 10:8-11 with v. 22 and the situation in 11:29-32). It is when the topic is resumed again that Paul mentions the existence of demons, and the fact that demons are the true recipients of the sacrifices (10:20-21). This notion is especially suggested by Ps. 106:37, in conjunction with 106:28, the text likely alluded to in Did. 6:3. The existence of demons do not mean that "the idol is real" (10:19), i.e. that so-called gods are real, because demons are not gods (v. 20, an allusion to Deut. 32:17). They do not have godhood as the Father alone has (8:6). Paul is not saying that demons do not exist (rather, it is the "god" veneer that is "nothing"). Paul thus saw the situation in the Christian church in terms of Israel's encounters with idolatry in the wilderness.
There still seems to be rub with Paul's "knowledge" premise: Paul presents the partaking of EIDWLOQUTON in ch. 10 in a black-and-white "cup of demons" vs. "cup of the Lord" fashion with no middle ground, yet there is a middle ground in ch. 8 which acknowledges that hypothetically a Christian who has the "knowledge" can eat with "freedom" without committing idolatry because of the "knowledge" that "so-called gods" are fake would block such eating as being considered worship. But in ch. 10, the implication seems to be that any eating of EIDWLOQUTON is idolatry. Thus it is striking in ch. 8 that idolatry is mentioned only as occurring when brothers with weak consciences partake of sacrificed food. So it looks like Paul's argument does shift between the digression, unless of course there is a better explanation!
(My reference to Flusser, btw, is to chapter 7 of The Didache: Its Jewish Sources and its Place in Early Judaism and Christianity, ed. by van de Sandt & Flusser (2002), pp. 260-265).