Export high bitrate video in lightworks
Depending on your definition of high quality, this is impossible.  For comparison sake, DV video (at standard definition) takes a gigabyte per 4.7 minutes, which is at a good, high quality level of compression.  You are trying to do 60fps 1080p video and get it under 2GB for 4 minutes.  That's over 12 times the information and you are trying to fit it in just over twice the space.

If you want to see how it works out when you use the highest possible quality that will fit your size requirements, you can simply calculate the data rate. Create a 2-pass VBR encoding with an average data rate determined by the size and time.  The file size is equal to the data rate times the number of seconds in the video.  2 pass VBR will ensure that the best possible use is made of that space.

You mention that you can break it up in to 4 minute sections to send over.  Why is it that you can't split the files (using file tools rather than video tools) and have them reassembled on the other side?  Is it a file size limitation or a file transfer limitation?  Also, is there a reason you can't send the original files along with the project files?  Each additional time that you encode, there is a slight bit of quality loss, so it is preferable to work from the original assets whenever possible.

Another possibility if you are working collaboratively and have issues with moving a huge amount of data.  It is possible to output low quality copies of the original video files and substitute those files for the high quality versions when working on them remotely.  Your friend would take your project files and the low quality assets, do any editing he wants, and then send the project files back to you.  You could then render the final output based on your friend's editing decisions but using the original, full quality assets.

The reason you don't want to go to a smaller file size in-between is that each time you encode a video, quality is lost.  The lower quality the source you are working from, the more is lost in each additional encoding, so if you move to a mid-range format early on, then the final output will suffer.  If you run the render from the originals however, you get maximum quality.