You can do this. The trick is to double click on the paint bucket tool and a dialog opens. In the dialog specify that the bucket only affects fills only by unchecking the paint strokes option.
Image 1: In live paint bucket options disable stroke filling
After doing this it works like you expected.
Image 2: And it works like expected. Even with a slight gap in the join of circle and diagonal thin line.
You can also copy the shape on a second layer and work behind the strokes if you wish to work with shape builder instead.
PS: Note that live paint bucket can live with small gaps but you can not adjust the size at least not in my version. But you can in shape builder so if you have bigger gaps then use that instead.