F5 APM and external mobile application log in

The most important thing to understand here is that APM's user interface is HTTP-based. There are certainly ways around that fact, but the standard interaction is HTTP. Therefore to consume an APM (form) logon process with a native mobile app, that app will have to talk HTTP to APM. The first thing I'd do is take a capture of the client side logon process to see what that looks like. At a minimum your native app should be able to follow HTTP redirects and consume HTTP cookies. You can get around the redirects, but you definitely need to be able to consume and use the cookie if you want to maintain any sort of statefulness.