I've seen a few videos of Andy Timmons playing that and others playing it as well and they utilize some tapping harmonics of some sort which I cannot replicate.
Is that tapping harmonics? How is it done?
From looking at Youtube videos of him playing it: yes, it's tapped harmonics. If you listen closely you'll notice that he also struggles to make them all sing (at least I noticed on the two videos I've looked at so far), so it's clear that they're not always easy to do.
I find a light touch to be important to make them sing, and it takes practice to figure out how to tap the strings without hitting the frets too hard because that will depend on how high the action on your guitar is. One of my guitars has a fairly high action, and I noticed they seem to be easier to do there because the strings don't hit the frets so easily.
It might be easiest to start with the octave (twelve frets above your fretted finger) before you try to tap other harmonics, and you'll need to be precise in where you tap. Since the underlying note of your harmonic is based on the fretted note you'll need to tap the fret of the harmonic above it, instead of where you'd usually put your fretting hand. So if your base is a D on the 5th string (5th fret), tap right on top of the 17th fret. Thus, your tapping finger position shifts slightly compared to tapping regular notes.
With a little practice you should find them fairly easy to do, good luck!