Depends on your FCP timeline codec. Why would you need a entire 4 minute file from After Effects? What you should be doing is treating only needed clips.
I personally export Animation files for one reason - super clean grafix with embedded Alpha Channels. It is the best codec for this use - file size be damned. If I don't need alpha channels and since I export cuts for treatment to a PC AE machine from a PC Avid, I use the Avid DNxHD codec @ 145 which is a much smaller file than Animation codec. I have these codecs installed on all my machines so I can read it on Mac & PC. FCP will see the Avid files...just set the timeline to the ProRes codec & render (pretty quick). Workflows between Mac & PC are ridiculous, but a necessary evil. Here is where you download the free Avid codec package (scroll to the bottom of the page to see the MAC & PC download files I listed below):
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avidtechnology.custhelp.com]
Avid Codecs LE 1.10 MAC.dmg
Avid Codecs LE 1.10 PC.zip
...and yes...this sounds elitist, but getting the PC peeps to work on AE on a Mac is the way to go. There are so many more choices of hi res codecs to choose from and the workflow is much more efficient IMHO. We are dumping our 4 PC Avids for Macs this month for the main reason that going back & forth between Mac & PC kills us time wise. The workflow is a log jam. Automatic Duck is a viable option as well.
When life gives you dilemmas...make dilemmanade.