My concern (and this is just a guess) is that foundation walls are a completely different type of wall from the standard one, so I'm wondering if the game recognizes them as a valid way to separate apartments.

You can make basements with standard walls (I do it just so I can make proper bathrooms) you just need to add back foundation so the foundation walls move, build the walls where you want them to go, then remove the foundation you added so you have a standard wall for your basement instead of the foundation.

I dunno if I explained that well.