Opinions vary here, as you see. There are at least three sets of stats that need to be checked: trailer gross weight vs TV max, which you have done and sounds good; TV payload weight vs weight of passengers, full fuel tank, tongue weight, and any luggage/cargo, which you do not mention and which is likely going to be the problem; and axle weights. There is also the matter of engine power vs wind resistance. A full height TT is like dragging an open parachute. It is there you will quickly find limits.
For what it's worth my TV is also rated 6,000 lbs and I drag a similar trailer around without issue, but mine is a diesel with twice the torque, and I wouldn't want to attempt it with less. TV payload is also where the pinch comes with my rig, and likely yours as well.
One other thing hasn't been mentioned yet: your transmission and gear selection. You are going to want to make use of lower gears on hills, both goig up and down, both to help prevent overheating on the climb and brake fade on the descents.
Good luck, and I hope it all works out for you.