Sorry, I Can't make out what your photo is trying to show. I have no idea what the specifics of your towbar is either.
What i looked for is an adaptor that would turn a swan neck towball into a flange type towball - this is it ( see previous link or direct here
http://www.bak-rak.com/s%20to%20f%20converter.htm ). I.e you can bolt your standard flange type ball ( or any special brackets) to this adaptor, and then fit this adaptor onto the swan neck ball - Hey Presto, you have a bog standard flange type towball on the back of your car.
It wouldn't be practical for towing something because your new ball would now be several inches too high but can't see why it wouldn't fit your bike carrier as it would be a bog standard flange.
Is it simply because you wouldn't be able to retract your hitch without unbolting the adaptor?