And thank you for your purchase of Sydney Pro.
Bistro theme is child theme of Sydney, not of Sydney Pro, so it will only work when you have Sydney theme installed.
Theme settings, menu and widget positions are saved per active theme, so by activating another theme or child theme, you will have to place widgets again, and to configure theme settings again. So you will need some time for doing this, but not too much. This is the case if you want to switch to Sydney Pro.
In case that you want to use Bistro theme which will use Sydney Pro as parent theme, then you have to change Bistro theme style.css file, or to create new child theme for example Bistro Pro. You can use existing Bistro theme files, and to copy them inside new folder (bistropro), and to change its style.css file so it contain Bistro Pro name, and it have to point to sydneypro folder so it can inherits its functionality http://screencast.com/t/3d8ijFv9K or simply create new child theme, and more about child themes can be found on WP Codex https://codex.wordpress.org/Child_Themes
Note that second step requires better understanding of child themes, and may require some coding knowledge.
In both cases, you have to set menu positions, widgets, and theme settings.