What red pen should I use to write on black paper?

You might consider a ultra fine point Sharpie in red.

Do you need large or small characters? I think a Sharpie would be a good choice too, but not ultrafine if you need larger letters. The chisel Sharpie is great for shaped bold writing—it’s like a big calligraphy pen. And they do come in red.

You might also consider outlining in silver or adding a drop shadow in black. Or both. Sharpies do come in nice metallic colors too.

Could you take a scrap of your paper into the art/stationery shop and try different pens? They often have testers.

Most inks are transparent, so you see the color of the paper through the line. That’s the case with conventional Sharpies and most other pens. So an ink line against a black background goes very dark, almost impossible to see.

Ideally, you want a pen that uses an opaque medium. These are usually sold as “paint pens” or “paint markers” (Sharpie also makes a paint marker). These have the disadvantage of being less user-friendly, in that they tend to clog and blob. It’s also harder to get a consistently fine line. All of this is because the pigments added to make the medium opaque also thicken it. But if you do manage to get it out of the pen and onto the paper, it will stand out better on a black ground.