I am not so certain he missed any thing since the chimney must be open
to operate the gas insert safely. You must have an incredibly strong
wind coming down the chimney to blow that fire out during full
operation mode.

Have the original guy come out and take a second look at his install
first. He may find something he overlooked that is casuing this problem
(ie. forgot to put the chimney cap back on, or didnt put the damper
back in properly)

If this is not the case you may want to consider a baffled chimney cap.
My father had an issue with this in his woodstove as wind was
deflecting off trees and forcing smoke to backdraft into the living
area of the home. A baffled chimney cap (looks like a puff ball
mushroom) from Lymance fixed it.

One thing to consider also is how much draft you fireplace allows in
when the fireplace is not in use. If this is an issue you may want to
consider a chimney balloon to seal the flue when the fireplace is
dormant. This is especially helpful if you find you have a tired old
damper.