Sure, I'm very familiar with the area. There are many rentals available in Bernards Twp/Bernardsville/Bedminster. What is your budget and how many bedrooms are you looking for? Since you need schools, I'm assuming you have at least one child and would need at least 2 bedrooms. Rentals start at around $1600 and go up from there.
Feel free to email me at firstname.lastname@example.org. I'd be happy to assist you.
All the best,