What quagmire would we (or another nation) "get dragged into" by presenting a ceasefire deal that you've already felt them both out on for approval behind the scenes before presenting it publicly? If they don't express interest you don't go public. You're simply giving them domestic political cover by being able to accept a deal presented by someone other than their opponent.
I'm not suggesting in any way that we be responsible for enforcing anything. Is it better to let this drag on indefinitely because neither wants to be seen as blinking first?