From the point of view of general relativity, inertial frames are those local reference frames which are in free-fall and non-rotating. "Local" reference frames because the free-fall condition may only be realised locally.
You pose the question referring to unknown sources \(S\) and \(S'\)of the forces acting on person and scale, respectively. If the force on the person is gravitation, e.g. if \(S\) is a distant star or a planet, then, prior to hitting the scale, the person is in free-fall (the person considered small enough compared with the scale of variation of the star's gravitational field for differences of the gravitational field across the person's extension, i.e. tidal forces, to be negligible). The person in free-fall, if non-rotating, is in an inertial system.
The scale cannot be shut off from gravity, but let's assume \(S'\) to be such that it overcompensates the gravitation on the scale. For example, a giant spring mounted on \(S\) could accelerate the scale away from the star/planet towards the person. Before making contact with the person, the scale is not in free-fall, and therefore not in an inertial system (it is accelerated relatively to a free-fall system, thus relatively to an inertial system).
After person and scale hit (and stick together instead of rebouncing) the person is at rest relative to the scale. The movement relative to the sources is a different question. As the scale still is pushed by the spring, the person now, via the scale, is also pushed by the spring. The person no longer is in free-fall, but accelerated relatively to a free-fall system; therefore, the person on the scale is not in an inertial system.
(By the way, something similar happens when you stand on the ground or actually step on a scale.)
Consider a different setup:
Assume you do your experiment in a laboratory which can be assumed in free-fall and non-rotating in the above sense. The laboratory frame is an inertial frame. Assume that person and scale are accelerated towards each other by springs, such that after making contact the person-scale system is at rest in the laboratory. Then, while person and scale are accelerated towards each other they are accelerated relatively to an inertial system and neither person nor scale are in an inertial system. Once person and scale have hit and are at rest in the laboratory frame they are in an inertial system.