Wow, that's a tough one Tasha. I know you're worried about Basil and I don't blame you. One of my three budgies is very timid and if I even drop a pencil on the floor when he doesn't expect it he panics.
One possibility that comes to mind is to try and de-sensitize her to noise. This is easier said than done but it just might work.
The first thing I would suggest is to make sure that when the work first starts that you are nearby to comfort her and tell her it's okay and that she's safe. Hopefully she will learn to associate you with the noise being no big deal when she see's that your not afraid and she will understand that there is no danger to her.
Also, try and spend as much time directly interacting with her as you can before the work starts if you don't already. I know that my sensitive budgie would rather just sit in a corner in his cage all day but if I let him do this he becomes afraid of everything. (I think this is learned behavior from when he was in the pet store- I got him when he was almost a year old. You may have the same challenge too)
You may have to become a little demanding with Basil. Although my budgie is afraid in the beginning within a very short time he's out of his "shell" when I don't just let him cower in a corner.
What is ironic about your situation is that parrots LOVE noise. Budgies are parrots and they do too. But not all noise. Try playing a radio loudly (increase the volume over several days) and try and get her used to the same station or talk show. I find my budgies like to listen to a male voice on the radio- possibly they associate it with me and safety when I'm not home. When the work starts, she may focus on the sound from the radio and not be as startled by the sound of the construction.
I'll keep thinking but I hope this helps.