Like I already told SSO is the best option even if you have Google, Facebook logins in original website.
For Android app
Simple solution is buy hosting at discourse.org. Discourse already have open source android which is currently work only their customers.
Else you have to autorize Discourse instance from android like how they did in their official app. Stll you may use SSO in this case.
I think since it is little advanced concept, if you want more help then you have to post a job in Marketplace. Then myself or someone else will help you to find alternative solution.