Is that even possible?

I´m not aware of any "video out" capability.

Use a HDMI splitter from, for example a AppleTV, and connect this to the Exclipse and Beamer. There are wireless HDMI options that work well. Use 'gaming-mode' and 'Picture-mute' to reduce lag and turn-off the LCD of the eclipse. Tada!

For limited lag and less splitters, I would recommend to split the hdmi coming out from the sound center : one HDMI to the LG screen and one HDMI to the beamer.

any suggestions of 4k splitters 

instead of splitting and beaming the video signal via HDMI to a projector, I am beaming the audio signal instead.

When splitting the HDMI between the soundcenter and LG panel, I realized I couldn't project the video of anything that is from a native app on the LG (such as Amazon Prime, Netflix). So I tried another route- beaming the audio signal since I figured it is also less data to beam. 

I have an eclipse 55, BL90, BL6000 with Wisa receivers, ultra short throw projector, apple tv3 with optical out. 

For the 5.1 sound, I send the optical out on the Apple TV3 to the eclipse (wirelessly via a bluetooth transmitter and receiver because my eclipse is in another part of the room, I only use it as a sound processor when using the projector and not as center speaker). I have the Apple TV directly connected to the projector for the video. 

The eclipse processes the surround sound and sends it to all the speakers via WISA. 

So far pretty good! the one issue in this setup is that I can't get Dolby TrueHD or DTS master audio because the optical connection can't carry that bandwidth. I'm not sure if the Eclipse can decode those anyways