By all means weigh it. We weighed our 13' Trillium
with some stuff in it, not everything we carry, and it was 1700#.
Here is a hint, if you do not require a certified copy of the weight, some scales will let you weigh your rig no charge. That is what happened to us. If we had requested a certified weight slip, it would have been $5.
Weigh and record the entire rig, then pull the tv off the scale and record the weight of the trailer. Then unhitch the ball of the trailer with the tongue jack on the scale and jack it up until it no longer places any weight on the tv and record that. Either weigh it empty or fully loaded. (I wish I had weighed it fully loaded. This way I am not completely sure how much we carry when loaded for the road..).
Subtract the weight of the trailer from the total and that gives you the tv weight. Subtract the weight hitched from the weight unhitched and this will give you the tongue weight.
Some people like to weigh each trailer wheel individually, which if you have the time and the scale isn't busy I guess you could do. Then you will know how to load the trailer side to side.