Sorry to reply so late. You will need to make some decisions because you have several challenges. So my advice is as follows:
1. If you live in NJ and work in NY, verify the income tax because you will be subject to income taxes for both states.
2. Commute to work - Working near Grand Central is the easy part because you can use public transportation (commuter rail) and be at work within 30 min. Your spouse who works in Purchase will have the stressful commute from NJ because of traffic on the bridges and highways. There is no direct public transportation from NJ to Purchase. All public transit from NJ goes to Manhattan and you need to switch to the MetroNorth to Purchase. Therefore your spouse is looking at driving 5 days/week. Time frame - from NJ to Purchase: anywhere from 45 min. to 2 hours with the average 1.5 hrs. The easy way would be taking the Garden State Parkway, Palisades Parkway, etc. to 287 in Rockland County, NY, then taking the Tappan Zee bridge to Westchester and taking 684 to Purchase. This method will reduce traffic congestion and get you to work faster. It may be a bit out of the way driving BUT you will get to work faster on most days. The other way taking the George Washington Bridge to either the Henry Hudson Parkway or Major Deegan (87) to Cross Westchester Expressway (287) will take much longer because you will be sitting in traffic in NJ before the George Washington Bridge and on the bridge itself. I hear the traffic reports every morning and they always announce anywhere from a 30 min. to 1 hour delay at the George Washington Bridge. And this is between 7:00 - 10:00AM.
3. I'm a former realtor and life long NYC area resident. If I were you, it would make your lives SO MUCH easier if you lived in Westchester County. You will save money on income taxes (you don't have to pay NJ income taxes), gas, bridge tolls (George Washington Bridge is $15.00/round trip), Tappan Zee Bridge is $8.00/round trip, car repairs - plenty of potholes driving from NJ - Purchase, you can sleep later, etc.
Places to live in Westchester for under $700K? - White Plains, New Rochelle, Mamaroneck, Dobbs Ferry, Tarrytown, north Yonkers, Cortlandt, Somers, Yorktown, Mount Kisco, Valhalla, etc.
I know property taxes are expensive BUT they do pay for our excellent public schools. Since you have a child, you will want your child to attend a top rated school and Westchester has plenty of top rated schools.
Your worst case scenario? At $700K, you may have to compromise on not affording a house in a wealthy area with one of the top rated schools. So you go 1 step down. The schools are still good but the housing is a bit cheaper. OR you could buy a condo instead of a house and have your child attend that top rated school in the wealthy area.