You've already flagged up the time that extra security for the TLV flight will chew up. Add to that a minimum of 30 minutes from gate to hotel, 20 mins back to the airport, 20 mins through security and you are now at 3hrs 50 mins in the hotel.
Now, add to that the fact the Christmas rush will be underway and you could lose another 30-60 mins getting through immigration and a hotel is unfortunately left looking like a crazy idea.
So there will presumably be the play area. Giraffe is a kid-friendly restaurant airside in T1. There are some good areas to watch the planes from and if you go over to the gate 5 area, that might be a bit quieter if you want to try and get some rest.
I know that hindsight is all well and good so this will be little comfort, but I have young kids and would have booked a shorter - or much longer - connection myself.