File size is determined by two things, duration and bitrate. If you want to make the size smaller, one of those things (or both) must change.
How does one change the bitrate?
What is it doing to the .mov that increases the file size by that much? The raw video file I received, which is 5:39s, is only 110mb.
What was the CODEC of the original MOV file,and what was the Bit-Rate?
What is the CODEC of the Exported file, and what is its Bit-Rate?
You will very likely find your answer there.