More wrinkles here: El Al are now saying that

-- the car seat has to be IATA approved to fix into the airline seat
Fine: that's what I expected and what I know -- but how do they check this? Is the sticker on the side which says the seat is "certified for use in motor vehicles and aircraft" good enough for them, or will they insist on specific language about IATA or FAA approval?

-- the phone rep also said that for safety reasons, it’s almost certain that they will ask me to hold the baby for take off and landing, even if he is in the car seat
This seems weird. Why would they do this? Isn't it safer to have the baby in the seat, if you choose to provide your own car seat?

I may need to email El Al with the details of the car seat to make sure this is all OK. Has anyone encountered this request before?

Update after calling them: their phone rep now even says that they don't sell "Infant in Seat" fares at all and I'd have to buy a child fare. It doesn't sound like the price is so different between the two fare types, but it seems strange that I was able to find a published fare with the "infant in seat" passenger type on ITA Matrix but El Al maintain that they don't have that fare type.