Just four?
As a hint, there are actually 24 or 26 scenarios, depending on which version of the puzzle is attempted. But it may be possible to reduce it to 12 or 13 scenarios and still fill the requirements, which Random Dan may have done.
I think I removed all the symmetric cases, although it still feels like I’m missing something.
I’ll run this through my test script to check it. Even if the script reports failed cases, it looks pretty close. I expect there should be at least one solution which can identify the fake but can’t determine whether it’s lighter or heavier… but at a glance, I’m not totally sure if you’ve found such a thing or not.