I know you said you don't want gel pens, but I looked at various white ink pens for about 20 minutes, and there are literally no ballpoint pens with white ink pens. Ballpoint pens are also bad for writing. They require a lot of pressure, and the ink can be weird.
Gel pens are really good because they have smooth lines, and the ink is opaque, so it's nice and bright on the paper. I'm an artist, so white is a very important color. White gel pens are good to have because they are really easy to control and can do small details.
I would recommend Sakura Gelly Roll or Uni-Ball Signo White Gel Pens. I know that you said you used a gel pen before, and it didn't work well, but that pen might have been a defective pen, damaged, or old. Try using some again to see what works.
Avoid using felt tip pens or paint pens when writing because they get used up fast and don't produce the bestcleanest lines. As for the black paper, lots of art and craft stores sell it, and you might also find a black paper notebook. Good luck with your pens!