When I made my ADR this past August I was able to get the direct line to Via Napoli when I needed to negotiate table ressies (you can't do more than 10 there on the reservation line). I was told that I was talking with the restaurant, and I believe it was--strong Italian accent and just slightly rude...
I was told by the dining reservation line that there is not a direct line available to call LTT (for an even larger group).

I was told by the reservations line that I should stop in at LTT before our visit to check about our request.