I don’t think coated tongue is necessarily a disease, or anything to do with MS or your medications - particularly if it’s not sore at all.
It’s usually benign, and of cosmetic concern only, although can be linked to bad breath. As you’ve found, you can scrape it off harmlessly. It’s possible to buy a tongue scraper especially for the purpose, but some people use an old toothbrush, or just the end of a teaspoon.
An antibacterial mouthwash may help, but I find those containing alcohol tend to make my mouth a bit sore if used regularly, as it’s a bit too harsh and dehydrating.
Ultradex (formally Retardex) is a rather expensive but different mouthwash that might be worth a try. It doesn’t contain alcohol, and it doesn’t contain strong flavourings like mint or aniseed to disguise your breath - but which can also be a bit harsh and irritating.
At first, it takes some getting used to, because the expectation is so strong that a mouthwash would be minty. It feels odd and almost a bit yucky to swoosh with something that hardly tastes of anything - you can hardly tell the difference from water.
It does come with a little sachet of mint that you can add if you prefer (a bit like salt and shake crisps, where you can choose whether to add the salt). I didn’t actually like it that much with the mint, so I think it was better tasting of nothing, if you can overcome your preconception that it has to taste like paintstripper to be doing anything.
I’m not saying it’s a cure, but it’s a different approach to mouthwash, that you may want to try if you’ve had no success with the big brands.
A change of diet, with more fresh veg and roughage can help too, but I think, to put it rather crudely, if you eat more fibrous, scratchy food, it gives your tongue a bit of a natural scrape. I don’t think it’s anything about the nutrients that miraculously cleans your tongue up, but just the physical properties of the food itself.
Anyway, a few ideas to try, but I don’t think it’s anything sinister - just annoying, and can make you self-conscious. Compared to MS itself, it’s very low on the worry list. But I know that minor issues can bug you more when you don’t feel well to start with.