By all means I respect wanting to roll your own version of this. But if you would like to buy something to detect vehicles from your rear, there is something that already exists and is used commonly by cyclist. You may however need a bike computer as well, so it might get a bit expensive...
You're absolutely right that that does look like it would do the job, but once I start throwing in a computer as well it may start getting a little bit prohibitively expensive as you suggest. Definitely something to keep in mind though, thanks!
Looks like you can get a dedicated display with the Varia for an additional $100. $300 total is still cheaper than a GoPro, Raspi, and all the additional stuff you will need to make it work together...
https://www.amazon.com/Garmin-Varia-Rearview-Radar-Light/dp/...