178 East 80th

Both not bad to suit your needs and way under budget.
I'd like to see you forgo the terrace and score a bigger kitchen and dining area.
Can you squeeze 2 kids together and do a three bedroom?
It would restore dining space that became a br in many cases and it would increase your search pool.

Terrace/balcony is not a must-have, but a very nice-to-have. Definitely want a 4-bedroom though, we have 2 kids now, but want 3-4. Maybe that's the question - are we underestimating how cramped this will be with 3-4 kids?

Alternative would be something like this:
http://streeteasy.com/nyc/sale/593087-coop-360-east-72nd-street-lenox-hill-new-york
(just as an example). Way more space, but more expensive, location is not as good, and PS 183 is good but is not PS6.

Talk about the combo from hell.
Some apartments were never ment to be together.
Screw the the forth bedroom and get a kitchen and dining room. Your house will be holiday central. If a forth kid shows up send it boarding school. I know a place in India with a work study program. (mostly sneaker manufacturing and sleeping)

maybe you copuld get the kid one of those disney internships??...cleaning toilets, while paying above-market rent--great resume item

The kitchen is really too small for a family of six. Also, are there really enough closets?

Your additional kids don't exist yet. If you do end up having them it will still be years before they are old enough to deserve their own room. Until then sharing a room will only make them closer and it won't kill them.