Call Facebook mobile native app for browser-based webapplication authentication

Hi! Our web-based app (not a mobile app) uses Facebook authentication for users to login. We provide a 'social wifi' solution where users can access WiFi by logging in with their Facebook account. Problem: 99% of our users are mobile users. When a user wants to connect with their Facebook account, they are redirected to the web-based Facebook login page. The problem is that the majority of the users don't have active sessions on Facebook in their browser, because they use the Facebook app. This means they have to enter their full username and password in order to authenticate. Desired solution: We would like to find a possebillity to bypass the browser-based Facebook authentication. And instead open the Facebook app to authenticate. I was wondering if it woud be possible to: 1) Open the native Facebook app as authentication method. This way, they don't have to enter their full credentials. 2) If the Facebook app is not detected on the mobile, then the user should login normally through the browser. I included a sketch that hopefully will make things a bit clear. Kind regards, Niek