Not sure why you need a HAT to do that...
The streaming program could be set up to auto start
, run at the touch of a button connected to a GPIO pin or maybe from a signal from the Pi Juice HAT?
Main concern I have is the Pi, video and WiFi are power hogs and the battery may not last long - play around with https://learn.pi-supply.com/battery-levels/
and feed the Pi 4 figures in (they only go up to a 3B).
Also there are threads on the forum about smaller distros (though not for the Pi 4 yet) and removing programs that are not needed - these may save power.