Interesting - I’ll give it a go in the bathroom!
Chas,
the walls around the windway can only affect very high frequencies, and are likely not the reason for what you experience.
What you hear you will hear with any instrument. A normal livingroom will be very much damped, especially in high fr. range, compared to a hard, reflective bathroom.
My Low D sounds “nicer” in my kitchen or bathroom than in my damped livingroom.
BUT, IMO it´s a matter of balance more than “support” of the walls. I played my Sterling Low D at my friends place a couple of weeks ago, best sound I´ve heard from it. Friend lives in a small house, only one room (35-40m2). I sat in the middle of the room and the room is highly damped BUT the ceiling is covered in plastic and therefore absorbtive in the low frequencies while reflective in the high range, giving a very nice tonal balance compared with the typical livingroom sound, plauged with standing waves and flutter echoes.
/Peter
The walls also affect the line-of-sight to the sound-producing part of the whistle. You’re right, of course, that any instrument will sound different in a small, bright room, and low whistles especially. But the shape and placement of the window also affect how a whistle sounds to the player relative to how it sounds to the audience. My Abell, with its short beak and long skinny window, sounds much different to me when I play it than if I record it or have my wife play it. OTOH with a Burke, and its long beak, this effect is less pronounced.
I find that the bathroom effect is much more pronounced with the Copelend low-D than with the Burke low-D. This could also be due to the complexity of the Copeland sound, too, I dunno. I was actually a tad disappointed with sound of the Copeland until I played it there, and now every time I take it out, I take it in the can for a few minutes.
Stop me if you’ve heard this before. I’ve told people in emails that I was once playing a slow air at a pub while everyone was packing up to go home. Someone turned around (toward me) and said, “Oh, I thought someone was playing chords with you, but it’s the overtones of the whistle.” (or something to that effect.)
Tony
One of the people you e-mailed that story to was me! 
Wha…?