set button background with image of window background

[gray]

Don't set a geometry, let it expand with whatever is placed inside.
I use grid, not pack but this is my choice, you should be able to make place work

a label can work, but a frame is better
add a frame on top of your root window, use relative width and height as follows:

self.background = Frame(root, relwidth=1.0, relheight=1.0, bd=0, padx=0, pady=0, image=self.background_image)
self.background .grid(row=0, column=0, sticky='nsew')

I'm not sure I have the syntax right as I haven't tested, but it's close if not correct.

If you switch to grid, please be aware that all widgets in the same container (the frame) must use grid as well.

Frame doesn't have 'image' option