It wasn't the threading, but the diameter of the threaded part. I tried another 15t freewheel on the hub, and it was also far too small.
If the reason that 16t were the smallest was engineering reasons, then it makes sense that they would decrease the size of the hub to get even smaller (15t) freewheels.
I should have known better though, when they had to look hard to find a 15t. Me and my picky ratios...