Hello there: getting into Lawrenceville as a day student is more difficult than getting into Andover or Exeter. The stats for accepted students from nearby WWP/Princeton area are top notch. Typically you need some extraordinary hook: valedictorian of your middle school, recruited athlete, URM or development case. My son attended Lville and he attended the local public schools before boarding in 9th grade.
I know a lot of his friends who were day students. They were top students at the local feeders (Princeton Charter, Stuart, PASH, Chapin, etc). A couple were Caroline Bradley scholars or had similar high level awards.
Your stats are fine (87% overall). I would send the scores in. Plenty of students with higher scores are rejected routinely.
Your parents seem to have the “Go Big or Go Home” mentality when it comes to boarding school. From first hand experience, I can tell you that it is not a good strategy. When it comes time to college applications, you will be at a disadvantage if you attend a HS where you are in the middle of your class. Better to go to a smaller BS where you can be at the top. Big fish in little pond > little fish in big pond.
Suggest you listen to some of the advice to add other schools to your list.