Michael, Coachman’s Whip wasn’t a tune familiar to me, but I’ve dug out the notation from the Fiddler’s Companion website and had a look at it. I don’t see any problem or difficulty whatever in venting the Ds in that pedal point passage (~E2 cE dEcE) you mention. I would finger it thus (ignoring the 1st beat E rolled crotchet):
c - oxo xxo,
E - xxx xxo,
d - oxx xxx
E - xxx xxo,
c - oxo xxo,
E - xxx xxo,
which doesn’t actually involve all that much finger movement at all, and no awkward alternations. If you leave the Eb key out of it it’s even simpler. On pedals and arpeggios involving Cnat to E I tend to use this slightly weaker C cross fingering (oxoxxo,) if the keyed fingering (x,oo ooo,) is not suitable as the best one on my flute (oxo xxx,) is rather awkward in context.
If you use oxx ooo(,) for C nat (noticeably flat and a little wolfy on my R&R) it is a little more finger-jumpy, but not massively so, and if you use the C nat key (if you have one) for all the Cs (which I wouldn’t, but tried just on principle) it is perfectly accessible with a little practice.
I have the vented D so thoroughly ingrained that it is perfectly natural and easy for me to vent it in (so far as I can tell, virtually) all contexts. I actually found it quite difficult (though it certainly involves still fewer finger changes) to leave L1 down for 2nd 8ve D in this passage, which I also tried on principle - in fact, I tended to hit a low D with it because my embouchure is so acclimatised to the vented fingering.
So, as usual, familiarity and practice have everything to do with such issues, plus some common sense about checking out what works best on your particular flute (N.B. I do not mean by that what you find easiest to finger as the player, but what produces the best sounds/the sounds you desire from the instrument, with due effort and practice).