The easiest thing for me is that second Isaiah calls out Cyrus (the Persian king who conquered Babylon and allowed the Hebrew captives to return to Jerusalem) by name. Either Isaiah was factually a future-seer (which is not really what the ancient term “prophet” means), or that part was written by somebody over a century and a half later (or Isaiah lived to be like 300 and took a break from writing for over a century). The only other way to salvage that Isaiah actually wrote all this would be to claim this part and only this part was a later addition for clarification. But 1) the entire context of that section (what we call Second Isaiah) fits very well together as a unit describing the social realities of post-exile, and 2) as soon as one admits there were later additions it’s pointless to view the entire author as Isaiah (as in, admitting later expansion is the whole ballgame).

If someone believes fully that Isaiah was a prophet of God who delivered God’s message accurately in the 8th century, then there’s not really a limit to what other kind of supernatural thing they might believe. With this kind of thing I don’t really try to convince anyone that I’m right, I only try to convince them to understand that as historians who cannot simply believe all the supernatural claims we see everywhere (as we cannot believe equally the claims made by the Hebrews, and the Assyrians, and the Greeks, etc), when we see an 8th century text explicitly switching to a 6th century context and mentioning rulers by name, we MUST, as historians, assert secondary authorship.
See: John Goldingay & David Payne, Isaiah 40-55 Volume I (The International Critical Commentary, 2006); Joseph Blenkinsopp, Isaiah 40-55 (Anchor Bible, 2000); Klaus Baltzer, Deutero-Isaiah (Hrmeneia, 2001).
It’s quite dated, but S. R. Driver gave a great summation of the evidence in his classic Introduction to the Literature of the Old Testament (Meridian, 1956), with special attention paid to the stylistic and structural difference:
“The prophet never abandons his own historical position, but speaks from it. So Jeremiah and Ezekiel, for instance, predict first the exile, then the restoration; both are contemplated by them as still future; both are viewed from the period in which they themselves live. In the present prophecy there is no prediction of exile; the exile is not announced as something still future; it is presupposed, and only the release from it is predicted. By analogy, therefore, the author will have lived in the situation which he thus presupposes, and to which he continually alludes” (p. 237).