I'm not totally up to speed on all the bridge camera options.
But the models I've seen, especially under $300, are pretty horrible. They may have some very basic manual controls, but most don't let you shoot RAW, etc.
Outside of the FZ200 and RX10 (both outside your price range), they have very slow lenses coupled with tiny sensors.
I may suggest something like the Panasonic lx7, though it may have a lot less zoom than you were looking for.
Amazon has a very good 1 day special on a refurbished Nikon J1 with 2 lenses for $289. It would fill your requirements of smaller, though still an interchangeable lens camera. Bigger sensor than super zooms. A fairly decent camera, though it lacks a lot of manual control.