Two best ways to do it.

1) Into a vocal or instrumental mic (SM-57 etc. on a mic stand: Use a windscreen. Play about 2" away and aim the mic between holes 3 and 4. Do NOT mic the fipple unless you are in a very noisy bar. You will get way too much breath noise, so you MUST use a quality wind foam. Turn the bass EQ off. Yes, off. Don't let anything below 200 Hz get through. Very important -- tell the guy running sound that whistles in the low register are softer than anything else he or she is micing. Whistles in the high register are louder than anything else ... and it varies as you go up and down the scale. So he or she should set up a very tight compressor, or leave it in the middle and YOU move closer and father away. Or both.

2) If you don't intend to play into a vocal or instrumental mic, get a lavalier mic, preferably with a phone jack or 3-pin professional jack. Put a windscreen on it and, place it below your fipple and hold it there with electrical tape or glue or a rubber band or velcro. Do NOT aim it right at the windway or it will sound like an Arctic gale. I go 45-90 degrees. Depends on the mic, the wind foam, and the whistle. Experiment ahead of time. Be very careful of the wire, as it will probably pick up noise from moving around. Same business with the bass EQ and compressor.

I don't see how reverb helps. It just muddies up your sound. But I won't argue with you about it. A lot depends on what else you are playing with, where you are playing, and whether you use a cheap whistle (shriek city!) or a Copeland or Burke (almost no shriek).

In short you cannot properly mic a whistle through a PA system, but what the hell. It's like a session - no one hears you on the low notes and no one likes you on the high notes.