For any other poor soul who runs into this issue:
I had the same issue - contacted
Blue Iris support (the free version, which basically means you can expect ~4 days for a reply for every email you send (as of Jan. 2022)).
After 3 weeks of back and forth with the support agent, he agreed with my diagnosis that the Pelco camera was shutting off the RTSP stream shortly after Blue Iris connects and the RTSP port could only be re-enabled by rebooting the camera and then sent me off to engineering. (Side note, the RTSP stream clearly works outside of Blue Iris as I was able to stream with VLC for over 30 hours straight with no issues.)
Engineering replied and basically said "Yup, that's what seems to be happening. You can either try turning off the option to send "keep alives", or contact Pelco for updated firmware."
I've tried turning off keep alives and am running the lastest firmware on the Pelco cameras (which are out of service life now), so I've concluded that either I will need different camera software, or different cameras.
TLDR: There's a good chance you can't connect older Pelco cameras to Blue Iris.