>I went to an AGU session on this a few years ago, and was surprised that there were still a number of competing explanations, with little apparent cross-dialogue between proponents of different hypotheses.

Interesting. Was there someone who calculated wether there is a solar irradiation difference between northern and southern hemissphere?

>I suspect the whole Arctic Sea is hot, but it looks like there’s a Barents hot spot because that’s where there happens to be open ocean.

No this spot seems to be very localized and it is basically always there (I haven't found any image where there is no norvegian front-hot spot , in particular here you can see that right east of it in front of Novaya semlya the water is also cold, this is not always the case but fairly often:

So this spot seems to be surrounded by cold waters (or ice or land).

this "cold" region in front of Novaya semlya is also ice free in January:

Sunil Vadakkepuliyambatta talked about tectonics in relation to methane leakages, I haven't seen his thesis but in fact
there seem to be some tectonic faults at what looks like as being exactly at that spot:

which are decribed in this document on Wikipedia as: Major active fault or fault zone, where nature, location or activity are unknown
(side remark: It actually looks as if this fault could be part of a fault which is a break between the western europe/African plate and the Eurasian/Arabian plate)

This points way more to some hot stuff coming from the seabad, like gas or even lava than some latitude dependent feature.