If you don't want to give up on the camera you've purchased, you might check for an alternate firmware on the GoPrawn Forums. I wouldn't be surprised if you find one to enable live output on the hdmi port.
Going waaay off on a tangent, because I have a few of these on my project shelf:
There are a few dozen variants of this camera manufactured by a handful of Shenzhen OEMs, all with the same basic design, swapping out the SoC and image sensor. They're dirt cheap, but most of the time they lie about recording capabilities, up-sampling a 720p capable sensor to 1080p, or 1080p to 4k.
Interestingly a number of them run Android under the hood (despite the gui looking nothing like Android), and offer up a root shell if you connect to them with ADB.
Even more interestingly, (at least on the Allwinner V3 variants I have) a v4l2 video interface is available once you have a shell, meaning you can inject a statically built copy of ffmpeg. Since many of the variants (even in the sub-$30 range) have WiFi, it's then possible to not only stream out the hdmi interface, but ship an RTMP stream directly from the camera to a streaming service or broadcast workstation.