Not a great solution, but if iOS itself is asking for your password pressing the home button when the dialog is up will be a nop instead of returning you to the home screen.
1: https://twitter.com/marcoarment/status/841359803192561665
1: https://www.reddit.com/r/privacy/comments/51wqhd/dropbox_fak...
See also NOOP, NOPR, etc.
Basically, if you press the home button; nothing happens.