DisiJW....Good observations. Edom and Judah had a long complicated relationship, but clearly there was no love between them in the 6th century. I'm impressed by the Obediah wording condemning Edom
13
You should not have entered the gate of my people in the day of his calamity;you should not have gloated over his disaster in the day of his calamity;you should not have looted his goods in the day of his calamity.
This certainly suggests an active role in the devastation.
For me this strengthens the trace evidence, ....I see 3 lines of evidence. The only first-hand account in Jeremiah omits the Temple in the Babylonian destruction and refers to the Temple as accepting sacrifices, (your comment addressed below) and the faulting the Edomites in 1 Esdras (and similarly Obediah).
The Kings abbreviated later retelling is almost certainly drawn primarily from Jeremiah, and the Chronicler was revising Kings. It would seem easiest to believe that Jeremiah was correct that the Babylonians had not burned the Temple, but the Edomites finished the job after he and the others left for Egypt. The later retellings having assumed the Babylonians were responsible or were simply abbreviating the overall episode.
The other proposal, that mention of the Post destruction Temple in Jeremiah, refer to the burned foundations is not impossible of course. The Talmud 6 centuries later interprets this verse in Jeremiah in just this way. However, there is no other evidence and appears to be rather, based upon this verse.
Ezra 2, as you mentioned referred to the mere site as the House of Yahweh. However chapt 3 makes explicit that this was a resumption of sacrifices when the foundation of the Temple had not yet been laid. IOW, the site was not, prior to this, being used for worship.
Like I said provocative, hardly conclusive.
It is in the end a sad story of neighboring nations with so much shared history, perhaps even the worship of the same god, being in the end mortal enemies.