Fuel load is ~50% of the total takeoff weight of a commercial jetliner, which is to say, even a rough inertial estimate could well be in the right ballpark.
Roughly, for each passenger there's an equivalent mass of fuel on the plane.
If you're working with thin margins that might not be enough, but if someone's shorted the craft half its fuel load, as was the case for AC143, I'd suspect this would be evident.
I mean the way we'd do it is a series of sawtooth climbs, since weight typically has a profound effect on climb performance.
Other than a nerd-sniping exercise I don't see the point of going down this rabbit hole, because on an aircraft with pax the pilot's job isn't to screw around playing Chuck Yeager and estimate fuel with test pilot techniques.
If they can't fly with confidence that they're safe, then the pilot's job is to put the fucking aircraft on the ground as soon as able.
I was thinking more of a secondary passive reading done by a computer in the background to catch the simple cases where the plane is unusually light for the entered distance.
