Duct fans are for dumbies who cannot size a duct system properly.
It sounds like someone just connected a duct in the closest and easiest spot in the existing duct system, instead of actually checking the duct system to possibly find an area in the ducts that could provide enough air without robbing to much air from one or two rooms.
If the system can handle the additional load, but the ducts cannot then there should have been problems with insufficient air previously.
I'd go with the mini split if I had the $$$$. Though zoning is a great choice when installed correctly, you can't beat the quietness of a mini split. Keep in mind that with zoning, when the addition is the ONLY zone calling, you will be trying to force ALL of the air into that zone therefore increasing the air VELOCITY which is NOISE. The you will have to set up a bypass damper to alleviate that...Then setup...discharge air sensor....blah blah blah... Mini splits are remote controlled too. I think their pretty cool....Check out http://www.mrslim.com
As for a duct booster fan.... who came up with the genius thought that if the system is already moving all of the air that it could through a duct, that it could be further increased by a low torque, 1/25 motor??? haha the duct fan is more of an impedance to air flow than anything... And why take away air from the rest of the house??? Do you turn off a light in the house before you turn another one on?
I'm usually a fan of zoning, but this isn't the greatest application for it, especially as a retrofit on an existing system. Sun rooms are tough because they are so different from the rest of the house in terms of solar load and insulation (the insulation value of a cutting edge high-tech window still can't touch that of a basic insulated 2x4 stud wall).
There's nothing wrong with mini-splits. They're quiet and efficient. In this type of application I generally recommend either a mini-split or a PTAC/PTHP (the through-the-wall units you see in most every American motel room). The latter isn't as quiet, efficient, or elegant as a minisplit, but they are cheap in comparison.