I would vote for the overnighter in Y. Babies do sleep comfortably in their car seats (really makes a difference) and if she does wake up, you wont have to deal with the wrath of your fellow business class passengers. It wont be a big deal if she's awake in business on a day flight.
I just wanted to address this comment;
One time, my wife asked one of the FAs if she could help her with the carry-on luggage and the FA says "it is against union policy...not in my job description".
I will admit that the comment was a bit crass but passengers are supposed to be able to handle the carry-on's they bring. I would say to them, go ahead and put them in the over head and then I would give them a push or hold other things... I've had passengers actually hand me their bags to put away. It is true that worker's comp can get sticky and they'll scold F/A's who are injured handling passenger bags.
I think she might have just been burned out with all the people dragging all sorts of things on board to avoid baggage charges and then expecting crews to put it all away for them.
Speaking of, when you fly the business seg, at least ask about an extra seat to put the car seat in (it can't hurt) but ask about stowing the car seat in the business class closet or overhead, if they have room. They still might insist you gate-check the seat. Just don't check it in as luggage at the counter. The airlines can be really rough on car seats which could compromise its safety when you use it afterwards, not to mention if your seat gets "rerouted"...