Don't be fooled by maps. Bridges and tunnels back up quickly, and trains are occasionally delayed. Expect your commute to be double whatever Waze is currently estimating, and some days you will be pleasantly surprised that it only takes that long.
Also, $4k a month for rent is not going to get you very much within that radius. You can find a 1 bedroom, maybe a small 2 bedroom flat, but I would budget for $5k if you definitely want a separate room for the baby.
$220k is a lot of money. Congratulations on your success. But if you work in NYC, it won't feel like a lot of money. On the other hand, living there is an experience unlike anywhere else in the world. You might love it, you might hate it, but at least you'll never have to wonder about it.