Ok, I am back, and here is my experience:

The outbound flight from TLV to EWR was awesome. We got row 16 JKL for ourselves, so my kid had his own seat. He was awake throughout the flight except for two hours that he slept (noon nap). Our luck was that we had another couple with a baby next to us (16 DEF) that was about the same age as my son, so they ended up playing together all the time...

On the connecting flight to SFO, again, we had the extra seat in our row, so my son slept between us all the flight. Five hours without any problems.

Flying from LAS to FLL through Houston, I got an upgrade and so did my wife and son (wife as a companion), so the FC seats and space were more than enough.

Flying back to TLV started in PBI, where again, we all got upgraded on the leg from PBI to EWR and it was great.

On the flight from EWR to TLV, the flight was packed and we had someone seating next to us in 16 ABC, but with some luck we managed to move to 17 DEF where we had the whole row, so again - my son could sleep between us.

To sum it all up, I got away with it... If I had to re-think the whole itinerary - I would have definitely bought him a seat, and that is what I will do next time he flies with me.