Honestly, I would probably use a phone camera in a cage. Add an inexpensive external mic (first priority) and a light (second priority).
The cage gives you expanded mounting options and a way to mount the mic and light. At your budget range that will give you much better performance over most cameras in that same range.
Example:
Phone Cage - US$20
Movo VXR10 - US$40
LED Light - US$15
GorillaPod Tripod - US$35
That's US$110 or about £82