There are several ways to measure tongue weight.
One is to get a a tongue weight scale like this:
Another is to use the "bathroom scale method":
But since you still also have to weigh the trailer itself, what I do is make two runs through a public scale. On the first run I hitch up and weigh the two tow vehicle axles with the trailer axle off the scale, the roll the trailer onto the scale and weigh the combined rig.
Then I disconnect the trailer and go back through the scale to get the weight the tow vehicle by itself. The difference between the combined rig from the first weighing and the second weighing is the trailer weight. The difference between the weight of the two tow vehicle axles from the first weighing and the second weighing is the tongue weight.
Often you can do this for free at your county waste transfer station. Go at a quiet time of day and ask nice.
In your case, because you have a very light tow vehicle, you should also compare the total combined rig weight to the MCGVWR of your TV, which should be on your door sticker. That is important because the tow rating of your vehicle is listed assuming there is nothing but a driver in it, which of course is not the real world.
So be sure to load up both the trailer and tow vehicle as you plan to actually use them, with all gear, batteries, propane, fuel, passengers, water, tools, etc etc. It adds up quick. I think you will find that unless you travel very light, and don't carry water in the trailer you will be pushing if not over your MCGVWR. Up to you how close to the limits you are willing to be, but don't go over, its illegal and you would be exposing yourself to personal liability if you have an accident.
As for load changes from trip to trip, once you get your gear sorted out the big variable is water weight. Water is very heavy. A full fresh water tank plus the water in your water heater, neither of which are included in the trailer spec numbers, is about 300 lbs. In a 178 if you fill with fresh water you will be lowering tongue weight because the fresh water tank is behind the axle. As that water gets used it will move forward into the black and gray tanks and increase the tongue weight. You need to keep the tongue weight above 10%, preferably 11-12% of total trailer weight to minimize sway, while also keeping it above the max hitch weight rating of your TV. That can be a challenge, you might not be able to carry any water at all with that tow vehicle.
You might want to load all your numbers into this nice online calculator to see where you are. It also shows what the benefit of a weight distribution hitch would be, assuming your TV manufacturer allows it.