We only have a 1-hour layover at Frankfurt, so I doubt we have time for immigration there.
You have it backwards. It should read
We only have a 1-hour layover at Frankfurt, so I doubt we have time for transfer, because security and immigration will eat up a lot of time.
You have no choice but pass through Immigration in Frankfurt. Depending on the time of arrival, security, and then Immigration, could take a lot of time, or be quick enough. I was there early this week, and landed at 07:00. A few Asian flights, so security was packed. Immigration was swift, but then again I have a French passport, so I used the automated gates.