It really depends on what the slime is. If it is bright green algae color, spraying the waterline with a bleach solution kills the biota, and it falls off. Tar remover takes care of oily stuff and tar. Try wiping once a week with a sponge and dishwashing detergent, that will clean a lot of grunge off if it isn't well established. If you have rust staining on it, oxalic acid will take that off, and you will want to wax the paint
afterwards. IMO, paint is really fragile, and you're lucky for it to last 10 yrs. Our boat is painted, and that's a huge job!
Good luck with it, I can sure understand not wanting to re-paint the stripe. Markpierce's solution would work, of course, but I think you like the aesthetic of the white stripe.