Alright, so I stretched the B string until it seemed stable and did a neck setup. (I follow John Carruthers' method to the letter for the setup.)
On the positive front, the B string sounds more focused, rattles and clacks less now. (Thanks to @dedpool1052 for the intuition!)
It is still noisier than the other 4 strings, but it doesn't sound like an alien string anymore. It lost a lot of what I would describe as clunky metallic overtones. So, when I play even unplugged, even though I hear more clack and rattle on the B than the others, it's just a whole lot less "in your face" and it blends more with the actual notes being played. It's to a level that I can live with.
Maybe it's normal that the B will be noisier, due to its weight and size. I'm unsure. And maybe it's to do with flatwounds. Again, it's the 1st time that I try them on a fiver so I have no point of comparison. I don't remember noticing anything of the sort when I had rounds on this bass, nor on my previous 5 string (an Ibanez EDA).
On the less positive front, I'm now experiencing fret buzz on the B string, on the higher frets (at the octave and higher). I have raised the saddle to help with that, but it's now a bit higher than I find comfortable. I'm not sure how I will deal with this issue. Some of the buzz occurs on the part of the neck that is bolted onto the body of the bass, so it's not likely that adjusting the truss rod further will be of any help.
When I visited my luthier friend, he measured the frets and none were sticking out or anything. So, I'm ruling this out for the moment.
Finally, still on the less positive front, although the B now behaves a lot better, I can't say I love its tone. Especially when I play open string, I find it sounds hollow.
I will try to play the bass in the next days and see if the string settles in some more, also try to tweak adjustments and see where I can get.
The set of EB Slinky strings is on the way. So I will eventually try those on and see. It may just be that I don't like flats on a fiver, or that this particular bass doesn't work well with them.