We have been using AKG C420 Micro Mic for the whistle and it's been working more than adequately well. So I guess it should fit nicely for the flute also - with the odoe it's not good, it only picks up the reed. I know it is quite widely being used for vocals also, especially 'wirelessly'.
Its sound is typically condenser-mic-like so some people would reject it for that, but again that's a matter of taste.
The price of it here in Finland is some 200 euros, including headset and power adapter (phantom or battery), but I guess you can get it cheaper practically anywhere else.
I suggest that you'd test - if only possible - more than one brand, not only for the sound but also for the comfortable 'sitting' of the headset.
AKS