How one approaches the δὲ in this context depends to a great extent upon the theological presuppositions one brings to the exegetical task.
From the comments and the question it appears that some have a approached the issue from a Catholic perspective, with how this deals with the Catholic sacrament of marriage.
Most Protestant scholars would argue that it is adversative in some way. For example, my Logos Lexham Greek tools suggest that it is adversative.
My exegetical summaries book suggests the following, which shows the division:
QUESTION—What is affirmed in the clause ἐγὼ δὲ λέγω εἰς Χριστὸν καὶ εἰς τὴν ἐκκλησίαν ‘but I speak about Christ and the church’?
This clause is added to make clear that the reference of τὸ μυστήριον τοῦτο ‘this mystery’ is to Christ and the church and not just to the institution of marriage [Alford,Calvin, Candlish, Eadie, Abbott]. Another commentator says this clause applies the words of Genesis to Christ and the church in a secondary manner, but not as its primary interpretation [New Century Bible Commentary]. The δέ ‘but’ is not simply explicative, but having a contrastive meaning, as though the writer supposed his phraseology might be interpreted in another and different way [Marcus Barth, The New International Dictionary of New Testament Theology, Eadie, Ellicott, Beare, Scott, Meyer, Bruce, Lincoln].
Glenn Graham, An Exegetical Summary of Ephesians, 2nd ed. (Dallas, TX: SIL International, 2008), 510.
Being a dispensationalist myself I would argue that there is a distinction between Israel and the Church, and Paul is using this phrase to reveal something that was previously hidden, (μυστήριον, mystery), namely that the Church is the bride of Christ. Paul does link this to the relationship between a man and his wife in verse 33, with the relationship between Christ and His church being a an example that the husband is to follow in regards to his wife.