I connected the RAK567 to the network (using 192.168.100.1/test.php, username test, password test) and changed the static IP.

On the RTSP video stream you have then on the router (rtsp://192.168.1.89/cam1/h264), it is for sure a FullHD stream.

NOTE: The video stream I got from the RAK567 is FullHD@60Hz (it is what my screen shows me) while in 192.168.100.1/test.php it shows "720p" and each time I switch it to 1080p, it switches back to 720p.

Can you double check that your RAK566 is sending a FullHD stream?
=> Connect to the Flylink WiFi
Connect to 192.168.100.100/test.php (test:test) and change Pipe0 settings. I am setting Pipe1 to the minimal (QVGA) as if I put both to maximal, I am losing frames.