iOS Twitter login without using device credentials?

I'm making an iOS app that is meant to be used by multiple people on the same iPad, so it can't use the integrated Twitter credentials from the iPad. In other words, users need to be able to click a Twitter icon and then log into Twitter, post their Tweet (and a picture), and then log out.

I'm having a bizarrely hard time finding something that can do this. The Twitter API wants me to use the device credentials, and not let people log in using different accounts.

Can anyone point me to a library or method that can achieve this?

Thanks for any help.