Yeah, some really great ideas there! Thanks for that Ubizmo & talasiga 
I did some ‘research’ of my own this morning, and my head is frazzled!!
Basically, I had a ‘spare’, very cheap ($20 from ebay) headjoint, that I doctored to accept a wooden whistle head.
My theory being that simply sticking a low whistle head on a boehm body won’t work, for the same reasons putting a parallel bore head (R+R, Pratten etc) won’t work. So, by inserting a shortened parabolic headjoint (minus the lip plate, and the hole taped up… nice ) into the afore mentioned low whistle head, it’s possible to get a considerably better sound.
Two drawbacks - I was using a low F head, not a low D or C head, so it wasn’t as powerful as it could be. However, the higher notes were not as screechy as I thought they might be. It just sounded like a really lush wooden low whistle (the the infinite benefits of keys!).
However, I don’t know if it was the fact that I was using a low F head, not a D. Or perhaps the fact that the parabolic insert was not all the way up to the chiff (or is it fipple) and the top of the inside of the whistle head, that meant the upper second octave was a little flat (note hugely, but annoyingly … not… quite… there…).
My guess, although I don’t have the tools and correct head at my disposal to know for sure, is that with a parabolic insert all the way up to the top of inside the whistle head , it’s possible to get both octaves and into the third octave, in tune.
I understand what has been said about the body of the flute being designed for a different purpose, and fine tuned etc… But actually, if one was to do this (parabolic insert all the way up the whistle head), effectively it’s producing extremely similar results? Of course, you can’t have the dynamic changes and subtle nuances that you can with a flute, but instead a more fundamental collection of frequencies (perhaps less ‘reedy’, but more ‘clear’).
Anyway, they are my thoughts at time of writing! Does anyone have a low D or C wooden whistle head with a ‘spare’ parabolic headjoint to try this on a boehm bore???
Calum