Videomaker – Learn video production and editing, camera reviews › Forums › Cameras and Camcorders › Consumer Camcorders › Suggestions on a Good Video Camera › A quick search of B&H
A quick search of B&H indicates that only the Canon VIXIA HF R82 Camcorder falls within your budget range. It would appear to have all the features and quality your project requires. Perhaps others will have more suggestions.
Could you do this with your iPhone or iPad? Quality would be just as good although you wouldn't be able to see yourself while shooting.