I like what you are doing, yuiop.
A fundamental epistemologic rule is that analogies (and all knowledge is based on analogies…) are created for a given practical purpose, to play a function. Hence they are valid in as much as such function so requires, but not beyond that boundary. The dogmatic approach forgets the function and tries to apply the letter of the analogy to the last consequences, no matter how absurd they end up being. That is the case of the Prince faced with Cinderella’s stepdaughter, who has trimmed her foot to fit it into the slipper, ie she fits into the letter but not in the spirit (the pragmatic function) of the concept. However, a dogmatic Prince would cry: I have to marry her, they are equivalent for all purposes, to all effects! And if Cinderella’s father happened to force his foot into the slipper by accident, the dogmatic scientist would still mount him on his horse, marry him and in the wedding bed, when the poor man protested “I am a man!”, our blind Prince would still insist (remember the final scene of “Some like it hot”) “Nobody is perfect…”.
Coming back to the point, I fully agree that all this requires some work of conceptual clarification. I am not the most qualified for this purpose, but let me make some clumsy tries.
There shoud be no problem with categorising the concept of inertial mass as a relative concept (just like time or length), as long as one understands what that means: different frames get different results about the mass of a body not because the body changes as a result of the observation (it will still have the same number of atoms) but only because the different frames, in the course of their respective measurements, interact differently with such body. (In particular, if the measurement involves the acceleration of the body, which in turn includes time and the latter is a frame-dependent concept…)
In turn, as you point out, active gravitational mass may be an invariant concept. The two concepts are thus equivalent for many purposes, but not in this particular respect.
However, I am not sure that this distinction (variant versus invariant) is so clear as it should be and I am especially not sure that the clarification derives from pure mathematics. Actually, if we say, for example, that proper time is invariant, because the proper frame measures it and the other frame “calculates” it through an equation, then we could also say the same about coordinate time: one frame measures it and the other calculates it through a formula. I think that what we try to say is rather that proper time reflects what has happened in fact: a clock has ticked a number of times (ie a number of events have happened) between two limiting events. That is reality and nobody can discuss it. It is not that the proper frame is privileged, it is just that it is better placed to obtain a direct calculation, a direct measurement in this particular case (the clock in question is at rest with it).
The same applies to your example. An object cannot become a black hole due to observation. Yes, of course. But not because active gravitational mass can be mathematically calculated from any frame. It is the other way round: that cannot be because it’d be absurd, it would clash with the principle of reality and, once this is settled, we rule that it is an invariant concept (if you wish) but *in the sense that* any frame other than the proper frame, if it wants to learn what has really happened, is forced to make calculations and seek the proper concept, which is the key to the solution.
Because otherwise, only on the basis of algebra and without the constraint of reality, could not I perfectly argue the following?: “No, you are mistaken, you think that you are not in a black hole, but make the appropriate calculations and learn that *you are* in such and such frame, but *you are not* in this or that frame and both things are true…”
But please proceed with your calculations, because apart from this small comment, I fully agree with them and think they are very illustrative.