Most whistles will play up to about G or A in the 3rd 8ve… if you really want/have to go there - it tends not to be very pleasant, at least if sustained. Of course, you need to learn the appropriate cross-fingerings (similar to those for Baroque flute, but somewhat variable from whistle to whistle), which should at least partially solve your sound-quality problem. Just pushing into the 3rd 8ve by very forced overblowing of the 1st & 2nd 8ve fingerings won’t work at all well. As a general rule, a narrower bore whistle will go there a bit easier and sweeter than a wide-bore one, and a conoid bore may do better than a cylinder bore, which might be an argument for a Clarkes (Original, not one of those dreadful Sweetone out-of-tune toys) or Shaw, but that also depends on the design of the fipple set-up - and those particular whistles (unless tweaked) tend to be too breathy and air-hungry to do a good job in this area. Generations usually go there quite readily and sweetly without being too shrill if you have a good one and find & use the right fingerings. Your Feadog should also be OK if it has a good head. Good high end whistles will usually get these notes quite readily.
Major caveat: you do have to develop good breath control and be able to “support” your air-column/air flow properly from the diaphragm with an open throat if you want to hit such notes reliably and consistently with a good sound - if you just blow harder and strain or puff at 'em, they’ll still squeak and crack or sound harsh even with the right fingerings.
Try these fingerings (where alternatives are given, experiment to see which is best in tune/clearest and most ready to speak on the particular whistle):
Top C natural oxo xxx or oxx xxo or xox xox
Top C# oxx xoo (rarely ooo ooo - but that is usually flat)
Top D oxx ooo or maybe oxx xxx
Top E xxo xxo or xxo oxx
Top F# xxx xox or xox xxx
Top G xox ooo or xox oox
Top A oxx xxo
You won’t really get above that even if your ears can stand it.
Alternatives are: “octave folding” - using your normal D whistle but bringing those very high notes down an octave by selectively transposing parts of the melody down as best possible to make it seem seamless and unobtrusive while maintaining the flow and shape of the melody; or looking at the tessitura (range from lowest to highest notes required) of the melody and then choosing a whistle in a different suitable key which will give you that range without needing to use the 3rd register.
P.S. I hadn’t checked out the YouTube link before writing the above. Just had a quick listen to it and busked along. So far as I noticed, he doesn’t go above top D, so you only need the high C# and D fingerings, nowt higher, nothing exotic… shouldn’t be a problem unless you have a duff whistle, but you may need to work on your breath control/support as I suggested.
Oh yes, pardon my hobby-horse riding here, but it ain’t a “song” - no words/vocals, see! It’s a TUNE!!! 