Export Composition Background in After Effects
I hope the person receiving your 'green screen' lower third doesn't ever find you in a dark alley. If it was me I'd be looking to bring the hurt. Really, this is not the way to do it. 

Firstly there are more space efficient codecs with alpha channels than the default Apple Animation, try proRes 4444 or the newer Cineform

If you really can't afford the bandwidth to send the video with alpha, render out the alpha channel as a separate movie and send them both. I'm old enough to remember that's the way things were done in the days when dinosaurs roamed the earth and when sending a Betacam SP videotape by courier was the highest bandwidth way of sending material. There's no such thing as an alpha channel on analogue tape. 

You can achieve that by adding an output module to the render (right click the output module in the render queue and choose add output module). Change the Channels dropdown to Alpha instead of RGB or RGB+. If you want, you can use a compressed codec to save space (or compress after the render, which is what I'd recommend). Just don't compress too much or you'll end up with blockiness in your alpha channel.