I have a Burke DBSBT…a fantastic whistle…BUT…I find the C natural is not quite so clear or precise as the rest of the notes…I use the two finger C natural…Has anyone else found this…I’m not sure just what it is…slightly out of tune…??? just not so precise…is there a different fingering I could use…is it an unusual problem…in every other respect the whistle is fantastic…Les. 
yah, oxxoooo is not good on Burkes,for the most part. its not just on yours. That’s why some go for the thumbhole.
I have an older D Burke, before the blacktip, and it doesn’t like to be crossfingered far as I can tell.
Then again, (Ihope my handicap won’t disqualify me from participating in this forum) I have never met a C natural crossfingering that I could live with on any whistle so far.
I take my chances with half-holing even though my hearing is none too great. You can always slide around it a little bit, work with the attack, cut it, etc. Now I’m in way over my head…
On the session D, you can get a decent C-natural with o x x | o o o ; it’ll be some sharp, but not unacceptably bad.
On the narrow bore D, try o x x |x o x . It works well enough on my brass pro NB.
–James
At faster speeds, I agree with peep about the oxx. The forked fingering is better but I have yet to work that into fast passages. In really fast passages, dear Joanie told me she sneaks C#s in.
If the oxxooo had an equal timbre to the other regular scale notes it would be closer to the just-tempered C natural, i think, and a good choice. But I think its thin and reedy. It sounds decent on Old Bush in A part, though I end up using halfhole in B part.
At the risk of boring some (I take that one often) I remind all of Breathnach’s assertion that a C natural is “halfway between a B and a D.” With recent experience, I think it has something to do with him bein’ a piper an’ all. 
And, having spent an hour or so slowin’ down and studying Paddy Canny on Poll Ha’Penny, I am still not sure whether that first C is a nat or sharp, or one of Breathnach’s conception. An oxx might work there.
I have the same model and I really don’t find it any worse than other whistle as far as C nat. being sharp when played OXX-OOO. It’s still a great whistle overall. I just reduce breath a tad and it works. The most accurate C natural that I’ve heard was on a Silkstone alloy.
I feel I have the same problem on my high D brass pro. I guess I’ll try these alternate fingerings on some of the slower things I’m working on. My problem is more that I can’t quite tell if it’s right in an actual tune unless it’s in the key of G. If it’s in that mixylodian mode (or whatever it is called) my ear just isn’t trained to it. And the quality of the tone is quite different and I think that makes it harder to hear. I will have to study over your answers more carefully but glad to see some solutions.
Instead of Cnat in fast passages, some Irish flute players use oxoooo, which gives you a C+, which is close to Breathnach’s ‘halfway between B and D’.
try it out - it makes reels especially sound like the old masters do (as long as eveything else is there!)
I also find the Burke a shade thin on oxxooo
good luck
b