Hog tunes has a windshield mount kit that is next to invisible and give you 4 1/2" speakers. Another option is to check ebay for a used Harley RoadTech riser mount stereo. They were specifically built for Heritages and RoadKing and sound surprisingly good. They can be gotten for less than your budget amount. I put in a picture of one installed. Not mine.
The far better solution is a Reckless Motorcycle audio fairing. It fits in place of the Road King windshield on it's mounts. On and off in 30 seconds after the intial installation. It's a few hundred more than your budget, but worth every penny.
Here's an example of a windshield mounted unit. Harley Roadking Rock Box and Windshield Radio Stereo CD Aux in Quick Release | eBay