Sumi-e ink is obviously water soluble. If fountain pen ink is not, then stay away from it with your good sumi-e brushes!!!! I have no idea if your fountain pen ink is, but there are plenty of inks that are definitely NOT water soluble.

I doubt if most black fountain pen inks would behave like sumi ink, since many are made of dyes that will separate into different colors when mixed with water. Sumi ink is usually made with carbon black pigment (not a dye).

One partial exception might be Higgins Fountain Pen India, which has super fine black pigment. There may be others, but then they wouldn’t have the glue binder that makes traditional China ink dry water-resistant on “rice” paper and silk, so again, they might behave differently from sumi/Chinese ink.

And I’m guessing that some of the fountain pen inks (dye-based) might stain your brushes.

Agree totally.
If the original poster is seeking an already mixed liquid ink for sumi-e, there are several manufacturers that produce those and you can buy them often wherever sumi-e sets are sold.
If the poster is trying for some specific effect, such as a color wash, then regular watercolors or again sumi-e specific colored liquid or solid inks will do the trick.
If the poster merely wants a cartridge based sumi-e brush effect, there are several lovely “brush pens” with cartridge ink I find at better art stores for this purpose.
Fountain pen inks are likely to stain the brushes, which seems to be the simplest answer to the basic question.