Relative weight

Basically, yes. If you assume that the scale is in an accelerating rocketship (to provide the acceleration needed to impart weight) the reading on the scale (assumed to be oriented so the pans are perpendicular to the direction of acceleration) will increase by a factor of gamma = sqrt(1-v^2/c^2).

Note that if the scale isn't oriented perpendicular to the direction of acceleration, the problem becomes a lot more complex.

 

The scale has its own frame - the main point of interest is the force on the scale in its frame, but it is possible to define the notion of force relativistically for any observer.

Relativistically, this is most conveniently handled by the concept of a 4-force. If we assume our mass is pointlike (or at least very small), we can find the 4-force by multiplying the mass m of our weight by the 4-acceleration.

The 4-acceleration is just the rate of change of the 4-velocity with respect to proper time.

See for instance http://en.wikipedia.org/wiki/Special_relativity#Force

Another way of saying this - the three-force is the rate of change of the momentum of the mass m with respect to coordinate time. The four-force is the rate of change of the energy-momentum 4-vector with respect to proper time. So the 4-force has one additional component (the rate of change of energy with respect to proper time), and is computed with respect to proper time rather than coordinate time.

The reason to use 4-forces rather than 3-forces is that they transform via the Lorentz transform. In fact, this is a property of any of the various 4-vectors that I've mentioned - all 4 vectors transform in an identical manner (that's their defining characteristic) - i.e. they transform via the Lorentz transform.