It's definitely hit-or-miss... Calling check-in counter works most of the time, but not always. I always had a sneaking suspicion that when it gets very busy they will find any excuse not to let you in.

As for the hand-luggage-only checkpoint, I tried to use it three weeks ago on an LX flight to Zurich (LX is listed on the website you posted) and was sent back to the main LX section for the interview.

I suspect if your ISR-passport holder and speak Hebrew, you will be able to talk sense into the 'selector'... Most of them are somewhat humane, even if they seem to have a tendency to act like apes. Alternatively, you could always come to the airport even earlier, and check-in when the BA desk is still open for the morning flight...