There is not that much flute content above 5000Hz, and that which there is is emitted directly from the embouchure, generated by the switching action of the jet. The “cut-off” frequency of the air resonance in a conical flute is in the order of 1.5KHz, so the harmonics start rolling off from there. So yes, any reasonable mic will be fine at the top end.
There is another issue to consider - the directional pattern of the mic.
Most lavalier mics are omnidirectional. You can pick them because they only have sound entry grille at the front. The big advantage is that they are relatively immune from handling noise, and less prone to popping.
You can get cardioid (unidirectional) mics. They tend to have a basket head, or a set of slots or holes around the outside, as sound has to be able to get also to the back of the mic capsule. Cancellation is used to generate the directional effect. The directional characteristics are helpful in reducing feedback.
Cardioids are “velocity” rather than “pressure” mics, and more prone to handling noise and popping. They also suffer “proximity effect”, a rise in the low end as the mic gets closer to the source of sound. Proximity effect is a pest for singers and guitars, making them sound woolly, and so stage mics have inbuilt filtering to combat it. But it can have some advantage for the flute, where the low notes can have less carrying power. Can give the flute more “body”. But very desirable to have a foam pop filter, as well as the in-built mesh one.
Ideally it would be good to test the available mics on the flute. Perhaps your local music shop will allow you to record through each of them, and you can take the recording away to contemplate? Remember to voice-ID the mics before playing.
But if that’s not an option, you could just get a not-too-expensive omni and rely on desk EQ to tweak the sound to your taste. But now you have to consider the question of phantom power - many lavalier style mics resent having 48V unexpectedly applied to their nether regions, and some desks don’t have individual controls on the availability of phantom. One in, all in - oops, sorry!
So talk to your sound person first. You’ll need to consider connectors too - the only reliable connector for mics is the XLR, and it’s probably bigger than the microphone!
Terry