How to reduce the intensity of garlic in a soup?

Actually, soup is the easiest kind of dish to 'fix' from overseasoning. All you need to do is add more of the neutral liquid you started with--in this case, I'd use veg or chicken stock, but you could use water as well. Add maybe a cup at a time, stir well and give it a few minutes for the flavors to equalize, then taste. Once the garlic flavor has mellowed to your liking, add the other seasonings (salt, pepper, herbs de Provence) until they're back up to where you want them. Dried herbs will take a bit longer to completely release their flavor, so keep that in mind.

If, at the end of the adjustment, you have too much broth, just ladle some out and freeze it for the next batch!

I've heard that slow cookers, because they use water as a temperature conductor, don't generate enough heat for many cooking applications, including browning, sweating, whatever. The garlic may taste raw because it didn't have the time or temp to cook the rawness out of it. This would be the same reason garlic, onions etc are often sweated or browned in a pan before liquid ingredients are added.

So, can it be fixed? Maybe, maybe not. You might find a hard boil will temper some of the rawness, as the garlic may get a bit more cooked (I suspect a cooker would not rise to a hard boil to prevent burning, since slow cookers aren't stirred). The amount tempering you get this way may or may not be enough for your tastes. You might try pressure cooking, though I don't know if that will make a big difference as opposed to stovetop boiling.

As far as I know, no method of heating the soup as a whole will get you much above what boiling water can already do, at least until your soup is no longer soupish.

If the garlic was in larger pieces, you could take them out, saute, and re-add them... though I suspect if the garlic was in such large pieces you'd have already tried taking it out. Maybe if you strain all the solids and saute them? you'd get more browning in the soup, and more breakdown into little skittery bits, cloudy and mushy and stuff, and probably more oil, all of which I don't mind but you might.

If you can't get this batch to mellow enough, then you can either dilute as SgtStens mentions, or discard and start over, or find someone who adores the bite of raw garlic in their soups.

I'm a big fan of black garlic, most of the raw garlic flavour is replaced by delicious and delicate flavours.

I wouldn't replace all the garlic in your soup by black garlic but maybe start with half and work from there. Personally, I've used it in a lot of different dishes and nearly always got great results.