So, I heard recently of Einstein's Box, where Einstein tried to give an account of quantum physics fail with a box whose mass could be measured, and a shutter to let out light, thus energy, thus mass... and the flaw turned out to be that the uncertainty could be bestowed on the needle on the scale, or on the position of the box atop the spring.
Let's try this: The box is floating in space. Two identical streams of bullets arc past it, timed so that they're always synchronized (i.e., each bullet has an "opposite" in the other stream). Since the arcs eventually intersect, let's say that the arc enters "above" the exit of the opposite stream, for a specific orientation, such that the pair of arcs can still be exactly opposite without having to worry about collisions. The details of the angles aren't entirely relevant once we know that this can be done. To simplify things further, let's make the box spherical.
The reason for the identical opposite streams is so that the box, no matter its mass, is always identically pulled toward each stream, and so remains stationary. Conveniently, the identical, equal, and opposite motion of the streams also means that they affect each other identically.
So, now, if the box loses mass, this can be observed by the un-tightening of the arcs of the streams, while the box's own position does not change. I'm not sure if this is complete, but it might be a step in the right direction. I also thought that it would be worthwhile to post this, so that others can examine this idea.