Are you using the mobile version of Google Docs? I haven't heard this complaint from teachers or students, so I've tried to re-create this issue and cannot.  I'm logged into my Google Aps, have a Google wordprocessing document open, move to a different tab and copy some text, go back to the doc and am able to paste without needing to re-login.  Did I miss something?  I'm using the mobile version of GDocs.
Because of the clunky nature of GoogleDocs in iOS world, we've been having kids paste things into notes or another wordprocessor (we purchased Office2HD...mixed reviews) instead of working directly in Google.

As an aside, we've found that tables and spreadsheets work better than basic wordprocessing on the iPads!

Rob Gervais
Director of Technology
Enosburgh Town School District
Enosburg, VT 05450
65 Dickenson Avenue
802-933-7777 x243