One might even go so far as to call it a black hole sun.

Isn't that based on article where the authors clearly state that it was just a mathematical exercise, and requires dark matter densities that were ludicrous-to-impossible?

It's something else. Same name, though.

I'd love to see simulations of what the inside of such an object would look like. Is it dense all the way down to the black hole, or does it take just a little bit of falling hydrogen to create enough "solar wind" to hold the rest of the gas up like a hollow shell around it?

Yes, maybe they just wanted something to call black hole sun.

The first time a little red dot was seen the astronomers were dumbfounded and could only think of one thing to say:

What the LRDs!?!?!