Some of the New Rochelle schools are pretty good and others tend to the average. At the higher grades, the schools are generally much larger than other nearby areas, but at the elementary level, it depends upon the neighborhood school. If the complex on Davenport is the one I am thinking of, there are mixed reviews about it floating around, this one is a very low address number on Davenport.
For your price point, you're not too out of range of a smaller apartment in a better school district if you don't mind a multi-family house, or something that is in a commercial district, either a second or third floor apartment above a shop, as opposed to a full service building.
Check out the Westchester-Putnam MLS for some leads.
MLS
You can search by school district and see what's available. I did a quick search to see what was in range in some districts and there are some decent options available. The districts are not categorized all that well on the search bar, however, so you will have to search through the list by town to get to most of them. Look for Town of Eastchester: Bronxville, Tuckahoe, and Eastchester schools; Town of Mamaroneck: Larchmont-Mamaroneck schools; Town of Greenburgh: Ardsley, Irvington, Dobbs Ferry schools; and Town of Pelham to see if there are any that you'd like.
All of these areas are pretty quick rides to Grand Central, 30-45 minutes, depending upon train line and time of day. Some of the properties in Mamaroneck might not be as desirable with a small child, but you need to search under the town to get to the Larchmont properties that might be interesting. Some in Mamaroneck are fine as well, but the larger complexes that are at a lower price point may not be attracting similar working professionals to the community.
If you look in Rye, it will be the town and there won't be much in that price range. However, I just checked the WPMLS and they had listed a 2BR with Rye Neck schools in the town for $1950. So, they do exist, but they are few and far between, owing to the nature of housing in Rye. Just steer clear of Port Chester schools in the Town of Rye. You want Blind Brook or Rye Neck, so if it has a Port Chester address, I'd not recommend it from a schools perspective.
Westchester is great for children, so I am sure you will be very happy when you find a place, as there's so much to do.
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