That's strange that the HyperDeck 8K Extreme does not accept square division quad split SDI input signals, since the Teranex SDI to HDMI 8K supports this format (in addition to 2SI).
Recording 4 independent video inputs is not supported on the HyperDeck 8K, although it has been requested by more than one person (of course, people requested the ability to record 4 HD streams on the HyperDeck Studio Pro too, but nothing ever came of that).
You might be able to accomplish what you want with two DeckLink 8K Pro cards in a single computer (one for recording and one for playback). I believe these cards do support quad split SDI inputs, although you may need a pretty fast machine to encode the video.
Another option would be to use 2 HyperDeck Studio 12G units. I believe if you genlock the HyperDecks to the same reference source and use a RS-422 remote controller with support for controlling multiple decks, you should be able to achieve synchronized playback (since both decks will start playing at the same time, and the frame timing will be determined by the reference source).