Joaniejd,
Although I have not traveled in first class, I have travelled with children many times. I vote for coach and request the bulkhead, if you are flying at night(which often times can not be booked until the airport). One direct flight with kids is much easier, and hopefully the baby will sleep. I only had my son sit on my lap one time and never again. It was only a 2 hour flight, but not worth it. We always bought the extra seat.

If it is a day flight, you may want the first class. Extra room would be great.
One other consideration is that the other 1st class passengers may not want to hear a crying baby overnight. Daytime I would not be as self conscious, but absolutely at night in first class.
Also, the more stopovers the more likely you may be delayed.

Have fun with whatever you choose.

Travelatte

While 1st class sounds appealing, I would opt instead to get a seat for the baby! 8 or 11 hours either one is a LONG time to hold the poor kid on your laps. That can't be comfortable for any of you.