Using "parallel" templates for the same content/databass

Hello. I originally posted this question to the forum for the Business Directory plugin but have realized that it may have more to do with handling the wordpress templates in general:

This is kind of weird to explain so I’m including a graphic. What I want to do is have my website display the Business Dir content in two ways. In addition to the normal website, I also want a touch screen kiosk application that accesses a different set of template files via a fullscreen browser. These templates will have a completely different design to complement touch screen use. Ive gotten as far as creating a custom template that is associated with a page at https://www.website.com/kiosk. There is no access to this page from the website. However, the kiosk box will load this page up via a fullscreen browser that is fixed on this url. From there the user can browse categories/excerpt lists/ and finally single listings(posts).

The part I’m stuck on is how to load the next page from the categories list with a custom template listing the excerpts or sub categories from the first custom template that I’ve created. The links for the first page load the standard template because of how the links are formed. See image at this link