I think you are complicating it more than it needs to be. You don’t really need a widget at all. You need just a shortcode that outputs the form with the select, and also the data if there has been something selected. That data would be either the first view or the second view, depending on the parameters that you embed in the links and your form. So the page the shortcode is on is controlling the whole thing. This way it is not limited to one use on a site, although you should probably include a define in the shortcode that you check, so that it can’t be used more than once per page.
The form would not need an action attribute, so it will be handled by the current page it is on. It would need to have the current value of which view is shown, and the links of the first view would need to have the value of the second view. That makes it have to use GET instead of POST, which looks messier in the address bar. So if you can change the first view to use buttons or something instead of links, you could use POST.