Isn’t there some guy who makes keyless boehm flutes? I think it’s Dave Swindler. It seems that he could make a D-flute with the boehm style stuff. I don’t see any contact info for him on the maker lists so, who knows if he still makes flutes or not.
Coincidentally, I have a flute by Sam here for attention at the moment that presented the challenges you mention. The owner (from Ireland, playing for 10 years) and I both found the bottom D was weak and the embouchure unreliable. You might find something of help in what I found.
Firstly, the embouchure was very crudely cut, almost as if the work on it had not been finished. (Hint - never rush your flutemaker?) There were aspects that I have found in the past contribute to an embouchure that is very “touchy”. Cleaning that up solved the touchy problem and helped the bottom end a bit, but not enough.
Secondly, an RTTA analysis showed that, while the 2nd octave was reasonably well tuned, the bottom octave was sharp at A and B, and about 40 cents flat of A at low D. A typical first-half 19th century response. Of particular concern was the difference between low and middle D’s - about 32 cents. That’s enough to unsynchronise the jet reinforcement which is fundamental to flute efficiency. I did some re-reaming to reduce the octave width over the RH region and bring up the low D. That substantially improved the D response. I still have more work to do on the flute but am confident it will be much improved.
So, adding to what has been said before, check that there are no leaks, but also take a close look at the embouchure cut (compare it to other flutes under a bright light) and tuning. While a new head or an embouchure recut will solve the embouchure problem, it won’t solve a leak or a mistuning issue. Do all your homework before spending any money!
On my 2002 Self Indulgent Flute Maker’s Tour, I found the closer I got to Ireland, the harder flutes were to play. Not to say that they were worse flutes, but they certainly needed more skill and experience with similar flutes to get good results from. Whether such effort repays itself with better performance is impossible for us to prove either way at this time. A fascinating area for a longitudinal study.
Terry