What if the Barents is warming because the slowdown of the AMOC - and probably other factors - causes it to rise more to the surface - hence the hotspot west of Svalbard - and thus instead of the hot Atlantic water sinking to the deepest parts of the Arctic, it stays high, and heats up the Barents?
This is an incredibly persistent misconception. The warm Atlantic currents are surface currents. They do not "rise more to the surface" since that's where they are to begin with.
And hot (or warm) water does not sink unless external factors make them sink, of which there are basically two that can operate in tandem:
1) The hot (warm) water cools down on the surface, and due to it's high salt content it ends up sinking.
2) The hot (warm) current meats freshwater and sinks in spite of being warmer, since the salt content makes it heavier.
But then comes the "AMOC is slowing down" hypothesis which is based on increased amounts of fresh water (meltwater from Greenland) diluting the hot (warm) Atlantic surface water and stopping it from sinking!
Since the sinking of the (cooled-down) surface current is thought to be a large driver behind the AMOC (perhaps constituting 1/3 of the total), when less cooled-down water sinks due to lower salinity, the AMOC loses power and starts slowing down.
Some people who seem to like posting pet theories here about the AMOC seem to lack the most basic understanding of how it works.