What I would have done in this situation is to tell the agent that I have important family business to deal with in Israel and can you kindly book me on any flight that will get me into TLV as soon as possible? I would then tell the agent that I would be happy to come back to the counter in 20 minutes while I get a coffee or sit down.
If there is no option with LY I would ask to be that my ticket be endorsed to Delta , British Airways, Lufthansa or any other carrier even if the class of service is different and again I would have politely emphasized the importance of my travels to Israel and then ensured my return date is confirmed too.
I would have done this all with a smile and kept cool. While I would be upset inside I would do my best to keep it cool.