Part of the problem arises when you isolate your tapping as a separate, discrete activity. Then you’re trying to do two things at once - tap, and play whistle.
Instead, approach it holistically. Remember, most ITM session tunes are dance music. As you play, you should feel the pulse not just in your foot, but in your entire body. And it makes you want to … dance. To move your legs, arms, shoulders, head. Get up out of your chair and bounce around a bit, or hop from foot to foot. Just watch kids respond to music. Like that.
Foot tapping is just vestigial dancing (or as fifenwhistle says, marching). If you truly feel the pulse, then foot tapping is what’s left when you’re dancing in your chair. And dancing while you play should be the most natural thing in the world.
It’s no surprise that some of the best tappers I know are also good dancers. My friend Ben Power (bepoq1 here) is a fine sean nós dancer, and he can execute complex heel-toe tapping with both feet while fluting. Another fiddler I know can manage full-on clogging while seated, with a well-placed board beneath his feet.
So don’t think of tapping as something separate from the music you’re whistling, but as a part of your total expression of the music. Not as something artificial to cause you to keep the beat, but as a natural result of feeling and playing.
In fact, because the whistle lacks the ability to pulse a strong rhythm the way that, say, fiddle or flute can do, the addition of deliberate tapping to your whistle playing can sometimes really enhance a performance (if not overdone, of course). Just think of the classic combination of whistle and bodhrán, or fife and drum, or pipe and tabor. There’s a reason that sound has caught the ears of people for hundreds of years.