Is $5,000 to install a dampener system a fair price?

I live in a single story home with central heating and air. The heat is a gas furnace (original to the house with a new blower motor) and the a/c is shown in the photo. In the Winter, I noticed the back bedroom barely had warm air, while the living room was very hot. Now, with the a/c on, the same bedroom gets barely any cool air and is very hot, while the rest of the house is ok. I've closed vents in other rooms, but the airflow increase is barely noticeable. A technician inspected everything, including the ductwork, and said everything is fine, but recommended a dampener system for $5K because the bedroom is far from the a/c. I've rented similar sized places and never had this issue. I'm unsure if this is necessary for a small house or if I'm being taken advantage of. I wonder if I might need a larger a/c or something else.