It is possible to use one account for your library and multiple for writing, but there are some known complications.
When using multiple google accounts, formatting can trigger an authentication error due the limits of the Google API we use to make changes to the document. These errors can generally be overcome by using our optional sidebar add-on to format citations when writing documents using another account.
When using the sidebar, take note of this image for another complication which you should keep in mind:
The Chrome extension - which is always linked to your Paperpile account - inserts citations via the overlay on the left. The Bernhardt paper is in my personal library and therefore shows up in the search results. The sidebar, however, only searches your personal library when the account editing the document and your Paperpile account match. In this case, I am editing the document with my work account, but use my personal email for my Paperpile library. Therefore, the sidebar does not display the results from my library.
Keep this in mind and you should be able to navigate writing documents from multiple Google accounts.
TL;DR: use the Chrome extension to insert citations (as normal), and use the sidebar to format them.