The contract lounge in TLV is called DAN there are 2 of them (3 min walk in between). Absolutely nothing in the least special about them. Complete waste of time. Very basic food and beer/wine. I think they take PP check online.

If you are adventourous you have more than enough time to catch the train to Tel Aviv (1st stop) and take a taxi to Jaffa Port. Have a wonderful meal there and head back. Would be best if air you could first recheck your bag. As there is no baggage storage at TLV. Sorry no idea if u can check in earlier. I doubt it

Tel Aviv is a very fun place to stay the night but expensive for ao short time value. Best food around.