Several issues:
Truck Payload:
What is the GVWR on the truck? The highest I found with a quick search is 6800lb.
If your measured weights are 6080lb and you are adding 350lbs with the family...that only leaves only 390lb for the hitch and anything else you put in the truck.
Tow rating:
A 7500lb GVWR trailer is well above the 6500lb tow rating. Keep in mind, battery, water, propane, are not included in the empty weight. It's surprisingly easy to add 1500-2000lb without realizing it.
Hitch weight: Assuming 12-15%, you looking at 900-1125lb hitch weight.
Will the engine blow up or will the rear axle break in half...probably not. But expect it to be a pig towing and get pushed around due to overloading...if you are lucky it will just be a white knuckle drive but it can also turn bad if you react poorly to the bad handling. If you use it a lot, expect more wear and tear related expenses.
As someone else mentioned, best to go off GVWR of the trailer and assume 15% hitch weight...then you can be pleasantly surprised if it comes in with lower weights.
Either look at smaller trailers or a bigger truck.
