We should be speaking of "a novelty device that weighs more than its mass would lead you to predict", that is to say, an object the measured weight of which does not correspond to what a different object of the same mass would weigh in the same location.
Measuring the force of gravitation between two objects definitely doesn't measure the mass of those objects without resorting to weight; the weight is the quantity you're measuring. Similarly, the fact that the center of gravity for a two-objects-attached-by-a-string system will lie closer to the massier object relies on the massier object also being heavier. If the massier object weighs less, why do you believe the center of gravity would still be closer to it?
> I understand 'weight' as how heavy something is within particular gravity field (ie on a bathroom scale on earth) whereas mass is independent of local gravity. The schemes I suggest measure mass without resort to weight.
Yes, those are the definitions of weight and mass. We can measure weight directly, because it's a force and we have tools to measure those. All of our methods of determining the mass of something, as far as I know, rely on the assumption that if you know the gravitational field at a point, all objects with the same mass would, if located at that point, have the same weight. The most common method of determining an object's mass involves measuring the gravitational attraction between the object and the earth (colloquially known as the object's "weight"), and then imputing a mass to it based on that weight.
In the spinning example, I could say that two objects attached by a string and set spinning around each other will spin around a point that balances the torque from each object (this might not be, strictly speaking, correct, but it's close enough that I think it's suggestive). But torque is defined by force, not mass -- if one of the objects gets heavier without becoming massier, that should draw the center of rotation closer to that object, shouldn't it?
Or phrased yet another way: if one of the two attached objects is heavier than it "should be" according to its "true mass", then the two-objects-and-a-string system will have the center of mass and the center of gravity in different places. Those terms are currently synonymous, but if we had a novelty object such as undersuit described they would be distinct. Is there any reason to believe that the two-objects-and-a-string system would, if spinning, rotate around the center of mass rather than the center of gravity?