We use the custom fields as part of our login to an assessment. 
We call the assessment with the fields prefilled for the candidate (no manual login needed). 
We are linking a timed assessment to a non-timed assessment.
We set up a call to the 2nd assessment from a branching question at the end of the first assessment in order to seamlessly pass the candidate to a non-timed assessment.

If the timed assessment times out before the candidate finishes answering, is there a way for it to pass the unique custom fields to the next assessment it calls on the return link (so it does not ask the candidate to log in)?

Yes, I believe if you set the “Application Integration” method to POST on both assessments and then put the link to the second assessment as the “Link After Assessment” for the first assessment, it should work the way you describe when the respondent clicks the Logout button.