Is the energy of momentum stored physically? While an object is moving, relativity will say it weighs more, especially so as it approaches light speed. 
The increase in energy is then easily sensed as an increase in mass
(Almost as a rock in your hand feels 'lighter' than the rock that falls upon your hand, before the energy of velocity is transferred and potential difference becomes zero.)
Is this increase in 'mass' measurable by our 'balance type' weight scale, or is it better off being considered an 'effect' of the potential difference between reference frames in which the mass appears momentarily heavier in relation to an equal mass with 'zero' velocity? Very curious. **Almost like 'electricity' where the measured ground connection is considered 'zero' voltage, although we know the ground has a charge by evidence of a magnetic field and telluric currents, we still consider this electrical state as a  'relative' 'zero', right? 
Same thing as momentum, at least as I currently perceive it; pure potential difference and totally relative, whether it's 30-90 volts or -30-30 volts, its the same 60v difference.
 A: It is, in principle, measurable by a balance scale. In practice, this is difficult to do, because how do you user your scale when the thing your trying to weigh with it is flying away at close to lightspeed? If, however, you confine the object to a circular track so that you can keep it positioned over the scale for an arbitrary length of time to make the measurement, then you can measure the difference between the mass of the object+track system at rest vs. the mass of the object+track when the object is moving at relativistic speed. 
A: It isn't possible to comment on your viewpoint that relativistic mass is "an effect of the potential difference between reference frames", because that statement is too vague to be either true or false.  In particular, I can't even tell whether you think that the relativistic equations involving mass are correct but can be expressed in an illuminating way using alternative equations, or whether you think that the relativistic equations are incorrect.  If you want an idea to become a part of physics instead of just vague philosophizing, you're going to need to define your terms precisely, and then express your ideas in terms of equations that express the proposed relationships between the carefully defined quantities, instead of just using vague sentences.
When you weigh an object on a scale, you aren't measuring mass directly, but rather measuring a quantity related to mass, namely the force on the scale, which according to Newton (but not Einstein) is related to mass by $F=ma$.  Energy and momentum are other quantities that are closely related to mass (indeed, Einstein says that mass and energy are equivalent), and it's more common for experiments that test special relativity's perspective of mass to involve measurements of energy and/or momentum, rather than force.  The Wikipedia article Tests of relativistic energy and momentum lists a number of experiments that have been performed that test whether special relativity's equations pertaining to energy and momentum are correct.  The "Undergraduate experiments" section of that article lists experiments that are even so easy to perform, that they can be conducted in a typical undergraduate laboratory.  The results of those experiments show that Einstein's equations are correct, and Newton's are incorrect.
