Are fence posts sturdy enough to support an addition of a wooden cover?
Vernacular architecture - what's the worst that could happen? 

Build it and see if it fails. 

If it fails, build it again stronger. Seems unlikley to kill anybody or ruin your house if failure happens.

Vernacularly sturdier - don't cantilever it, add two more posts on the open side. Or let it fail, if it's going to, then do that when rebuilding.