I've been in contact with Accsoon, the manufacturer of the CineEye 2 Pro and 2S Pro (SDI and HDMI version) and I'm updating here in case anyone else with a similar issue comes across this thread.
I had similar issues with an Ultrastudio HD Mini as well as a Decklink Duo 2 - sending 1080p (50) over SDI from the CineEye 2S Pro gives a flashing black screen but sound gets through. It does not matter if the input signal is HDMI or SDI on the CineEye's transmitter. It also does not appear to matter what camera or settings is used as the source. The CineEye 2S Pro receiver side SDI output is where the problem occurs. Updating the CineEye to the latest firmware (that claims to support Level B SDI) does not resolve the issue.
Switching my camera (and thus the CineEye) to 1080i (50) solves the issue, as does 1080p (25).
Taking the SDI signal to certain other devices including a Feelworld P7S monitor with the exact same cables seems to work just fine.
Accsoon support says:
Decklinks need SDi Payload ID to recognise the 1080p signal. This is not a Level A or Level B issue.
CineView add SDI Payload ID so it can work with Decklink.
CineView refers to the newer model of transmitter/receiver that is replacing the discontinued CineEye 2S Pro.
I asked if CineView receivers/transmitters are compatible with CineEye 2s, and they replied no.
I also asked if the CineEye 2S Pro has an upcoming firmware update that adds SDI Payload ID to enable it to work with Decklink, and they replied "There are some limitations, we can't update it".
As the CineEye 2 line of products has been discontinued due to a parts shortage and replaced with CineView, this issue is unlikely to be of any priority to either Accsoon or Blackmagic.
I think the reason why it works with some other devices like the ATEM Television Studio HD switcher and my other monitors and devices is because those devices infer the format of the incoming SDI signal without referring to the Payload ID.
Hope that helps.