I crumbled and ordered a Tourtech Pro case.
Made some enquiries about them, and whether a 4003 would fit, and took a chance.
Finally arrived (story in itself! 😀), the 4003 and the Walnut Faker fit nicely, though there is a little bit of space between the lower bout and sides of the case, though it's not a problem when you close the case, just need to be careful when putting the bass away and make sure the neck is sitting in the channel properly. Overall, the case seems sturdy enough, light, comfy handle, and the clasps are cool, though there's only 1 with a lock. How it would withstand being thrown about as luggage, I don't know.
The Thomann ABS case is a little bit narrower.