I think there are some differences among these schools that may ease your family’s decision. For example, does your family want complete Harkness, as opposed to a mixed method of instruction? If so, then Lawrenceville would be the choice. However, if you don’t, then you might want to eliminate Lawrenceville from consideration.
Meanwhile, as my own family faces an even broader decision, I have spent considerable time examining the course catalogs of various schools, including Choate, Hotchkiss and Andover. The latter school’s curriculum in the social sciences beyond the history requirements is very reflective of the intellectual fashions of the day and the current social upheaval in this country. That of Choate and Hotchkiss are more traditional and less reflective of the specific intellectual interests of their respective faculties.
Meanwhile, from what I can tell I think when comparing just Choate and Hotchkiss to each other it comes down mainly to non-academic factors. From what I can tell from talking to folks, Hotchkiss is more isolated and the kids are more “scaffolded,” or watched over. The kids at Choate might be given more freedom, which can be desirable or not, depending on your kid.