First of all, CONGRATULATIONS for all of the important events you listed. Big year!
You've hit classic FT response territory, i.e in asking how to get a cheapish redemption you're being bundled off into the premium cabins, which I'm guessing are beyond the budget?
Personally in your situation I'd look at booking a cheap economy fare and maxing out the part pay with avios. It's not a very efficient way of using Avios for the purist, but if you're going to be tightening the purse strings with the baby and house for a bit, you won't be losing a lot by burning them. Also you'll get a chunk of avios back.
Also worth looking at BA holidays, helpful for cashflow because you can pay a deposit and the rest later, and they do have some good deals. I don't think you can part pay with Avios though, might be wrong about that.
I wouldn't be stretching myself to get into a premium cabin. The New York flight is fairly short, and you can do a lot more with the difference in price (even to a cheapish redemption fare in J) when you're actually in New York. You get the lounge benefits on status anyway.