FeatherHarp,
I know what you mean about fighting noise with noise. I have tinitis in one ear, so sleep with a soundmaker that sounds like crickets chirping all night. Makes it feel like summertime all winter long, and distracts from the damn noise in my head. Good idea. I will ask my sister to leave the tv on to a music channel, as it's pretty near the cage. It may be enough to help.
We do often leave it on anyway, just to keep the dog company, since my dog passed away. The lone dog was very stressed out when her life-long roommate "disappeared". She developed a serious twitch, a couple of days after Tody died. And it didn't stop for about a month. I mean continuous! Poor thing! It gradually stopped, and she has adjusted to being an only dog. At 13 she mostly sleeps all day. But the TV on, and my sister being there, may be enough to keep everyone calm.
The birdcage will "just" squeek into the bedroom, if I move the dressers, so it's a very big deal to move it. First I have to move a hall table out of the way, then take drawers out of the dressers so they aren't too heavy to be moved. I've only done it twice - when I had to catch the birds to have their nails trimmed. They were a bit overgrown when I got them. Now that they are trimmed up, and have their cement perch, I hope not to have to do that again any time soon! And in fact the bell is closer to the bedroom door, than to the birdcage in the livingroom.
Gouldygirl,
Thanks, I would think that anyone who can put up with crying kids is pretty much immune to noise! I don't have kids, as you can probably guess.

If yours are not too stressed out by construction noise, there may be hope for my brood yet.