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In your price point I would try a Lumix FZ80 which can shoot 4K segments of up to 8 minutes at 30fps, 100Mbps, has center point continuous autofocus, and allows you to set iso, aperture and shutter speed manually if you wish. It has a 20mm to 1200mm fixed lens. The FZ80 is a little over $200 used on ebay, and $300 new.
You also would benefit from a decent video tripod with a solid base and smooth panning like this one, which I bought recently for a heavier camera. It cost $124 on Amazon.
It is the “Regetek Professional Video Camera Tripod System, 65 Inch Heavy Duty Aluminum Adjustable Tripod Stand with Fluid Pan Head and Carry Bag”.