Update: I tested this again, but I turned off all audio enhancements I could find in Windows 11 settings, and I used my highest quality microphone. What I found is that on my Goldie whistle, the second octave can be played in tune with some notes being significantly LESS than twice the volume of the first octave; but with other notes, the disparity is greater.
Second-octave G and F# can both be played only 1.5 decibels louder than second-octave G (yes, you read that right!). Both could be blown in tune roughly 1.4x louder in their second octaves.
Second-octave D and E can be played in tune about 2 or 3 decibels louder than their first-octave counterparts. Both could be blown in tune roughly 1.5x-1.8x louder in their second octaves, depending on how warmed up I was (Goldie low notes get louder as you warm up).
The biggest disparities were in high A and high B. High A was 5 decibels louder, and 2.5x louder, in its second octave. B was 7 decibels louder, and a whopping 5x louder, in its second octave. (I always knew there was a reason I instinctually tend to jump down the octave when I play that high 
).
Of course, this is not a very scientific experiment. Decibel measurement can vary depending on how close you are to the microphone. Also, playing as loud as possible in the first octave and as quietly as possible in the second (while staying roughly in tune) are doable for a test like this, but not really feasible during real-world playing.
Still, I found it interesting that the range of the whistle most commonly used for Irish music (D to high A) can be played less than twice as loud in the second octave than the first, with the only exception being high A, which was 2.5 times louder. I imagine it’s very difficult to make loud, large-bore whistles with volume this relatively consistent. I tried several other whistles, and none of them were this consistent (Mazur seemed like the closest). I guess that taper in the head, combined with a lot of undercutting, is what accounts for the consistency.
I should also add that this Goldie is the second of two that I’ve owned, and the first one I had was even more consistent in volume (if I recall correctly) than this one is. Its second octave was actually so quiet that it bothered me. It almost felt like I could get more volume out of lower-octave notes than higher-octave notes (I know this wasn’t actually the case, but that’s what it felt like to play it).
Or record on a recorder with compression turned off and publish the recording here 