In general, you should expect to pay a premium for a rental that accepts pets, especially dogs, and among places that accept dogs, many will limit you to dogs under 20 pounds.
You're going to have some tough sledding here. Tonight, I see six homes in the Bellmore-Merrick HS district for rent that are both 3 bedrooms and pet friendly. 3 are in Calhoun's area, 2 in Kennedy, and the one Mepham home is cat-only. It's listed at $2800/mo, and none of the others dip under $2500/mo.