TOPIC:

How to export an After Effects template to transfer/use in Final Cut Pro - FCPX?

Not possible, AFAIK. You'll need to recreate it from scratch in Motion if you want to use it in FCP.

Ok. Assuming there is limited options, would an entire file exported out of After Effects as a .mov file with transparency help? Or suggestions? This motion part of the video (.mov is Adobe After Effects) is requested to be combined with a speaker video in Final cut pro. The .mov with transparency area layered above a speaker video layer so the speaker video shows through. And different named speakers (assuming the lower third is in Final Cut). And there is some background music in the .mov within the beginning and end of the banner.

Since I don't know what the template generates, it's difficult to parse what you've got vs. what you want.

You can export your .mov from AE as ProRes 4444 (make sure to select RGB + Alpha!) Then you could put that over your presenter, layer the lower third on top of that, add your music track and it's good. If you need the text/lower third baked into the AE .mov output, you could certainly add that in AE and export separate 4444 files for each presenter.