Small, and I mean SMALL file

If you want consumer files (h.264 MPEG) you'll have to export two of them. One rendering alpha only so you end up with a black and white file you can use as a track matte, and the other rendering the video. MPEG does not support 4 channels, only 8 bit RGB. You need a professional format that supports at least RGBA, has all of the information in every frame instead of making up at least 1/3 of the frames by predicting how the image is changing using the IBP frame sequence. Like Dave said. Storage is cheap, buy a drive.

Get more storage.  It's cheap.

What's the deal with filesize?

You can only export the lower third part, not the whole HD image. This requires prober placing in PR, or whereever it is inserted. 

If you don't mind quality, just export it as H264 (with a seperate alpha mask, since H264 has no alpha). You can shrink file size a lot, but don't expect a crystal clear image or accurate colors. H264 is only usable for final internet delivery. If there is any processing step like color correction, this is not an professional option at all.

However, if you want quality, there is no way to shrink the file size. Working with professional codecs like DNxHD or ProRes usual end up in very large files. The image information has to go somewhere.You can try to zip the files after export in case you have to transfer them.

*Martin