I have 1Password for iPhone and copy the generated password from there every time I need it. Certainly a pain but worth it. In the end, it doesn't take that long: 1Password is on my home screen, the Apple login is favorited and I can type my master password fairly fast. The most annoying part really is switching between apps.

I also have my phone set up for iOS not to ask for the password for 5 or 15 minutes (can't remember the exact option) after I entered it.