You should be able to weigh the tongue of a trailer that small with a simple bathroom scale.
It can be calculated, but you would need to know a whole bunch of other things before you could get anything other than a wild guess.
If you had:
1. Front and rear axle weights of the golf cart.
2. Axle weight of the trailer.
3. Distance from the hitch coupler to the axle of the trailer.
4. Distance from the hitch coupler to the front axle of the golf cart.
5. Distance from the front axle of the golf cart to the rear axle of the golf cart.
You could calculate tongue weight...
Another way to do it:
1. Distance from the front of the golf car to its center of gravity.
2. Distance from the hitch coupler to the trailer's center of gravity.
3. Distance from the hitch coupler to the front of the golf cart.
It's just easier to wait until your wife isn't looking, sneak into the bathroom, grab the scale, and weigh it.
Just make sure it's not one of those talking scales... It'll probably say "Get off me you cow!"