I went 14' between posts with this "sandwich". In 7 years I didn't have any sag to it. It was holding up an oversized roof too.

I'd build it without the middle post. If you ever noticed sag in the future you could alway jack it back straight and add the middle post at that time.

This is not a professional opinion by any means. :)

You should be fine with using 2x12's spanning that distance. I was doing some research on designing a deck and I came across a handy guide at:http://www.richmond.ky.us/Departments/Codes/Forms/Brochures/
Deck%20Construction%20Guide.pdf
It lists maximum spans and spacing for various sized lumber.
It says a 2x12 can span up to 18'10" if used as a joist and I think your purpose is as good as if using one joist.