Livingston seems to be a great fit for you and your family. This town offers a variety of what you are looking for if not everything on your list of "wants". A safe area, orthodox and non orthodox Jewish schooling, public transport by bus and train (all nearby) and a Jewish population second to none in Essex County which will not let your children return to Israel as "foreigners".
I chose to live in this town for similar reasons about 6 years ago. My happiness today is evident of my choice. My daughter attended 3 different schoold from Beth Shalom to Jewish Montesori to the JCC (Jewish Community Center). All have provided a top notch education and have molded my daughter for the better. We could not have been happier with those decisions as well.
I welcome you to Livingston and extend my professional services as a Realtor to you and your family. Please do not hesitate to contact me for any real estate related questions or local schools for that matter. Best of luck to you and your family in your search.