The premise is already wrong. Any kind of copy mechanism will be imperfect. Even if we just consider classical physics (which is obviously a very coarse approximation), there is no mechanism that could scan every atom of your body at precisely the same time. Invariably, there will be an intervall ("all atoms have been scanned between t0 and t1"). Now relatively basic math tells us that the development of a complex dynamic system can change arbitrarily with small changes in the starting parameters. Maybe there could be an argument about the statistical behavior of the system (for instance, it is very unlikely that the copy would immediately lose an arm or a leg), but the state of mind would definitely not be identical.
If a body is frozen to low enough temperatures, then the movements of all atoms are reduced to vibrations around some fixed average positions.
After freezing, in vacuum, you could remove the atoms from known positions by shooting ions to them and then you could identify the removed atoms.
Of course this would be exceedingly difficult to do for a large body, because identifying the type and positions of all atoms would take a huge amount of time and because if the body is large it is difficult to freeze its deeper interior fast enough to prevent any damage.
Nevertheless, there is no theoretical reason for this to be impossible.
But you wouldn't measure the living body. You would measure a frozen (that is, dead) one.
And even then your level of precision wouldn't be exact but subject to errors. Regardless how small your measurement errors might be, there cannot be such a thing as an identical copy.
Einstein's elevator metaphor did not at all draw any important bits from the impossible construction. A spaceship under constant acceleration works just as well.
A thought experiment that creates a problem out of an impossible premise is just worthless as such. First of all, everything can follow out of a wrong premise, and secondly you can create an arbitrary amount of such experiments, preventing any insights.
Classically you should be able to measure any "information" present in the system. In general a useful machine cannot depend on indistinguishable microstates. If you include quantum mechanics then you may run up against to No-Cloning theorem, but that's a contentious question.
