You don't need or want 20% of the trailer weight as tongue weight. 10-15% is normal and adequate.
If you have 700 lb of cargo capacity left after family & weight distribution hitch, that gives you a loaded trailer weight of 4700-7000 lb. Allow about 800 lb for trailer contents, and that puts you looking at trailers with a dry weight of 3900-6200 lb.
Given that your family is going to grow and weigh more....your available cargo capacity is going to shrink. So definitely keep the trailer on the light side.
I have no suggestions about trailers but most are designed to sleep 4-5 people max. Good luck. I hope you find something and are able to have some great times camping with your family!
Single empty-nester in Middle TN, sometimes with a friend or grandchild on board