You seem to conflate Copenhagen interpretation with objective collapse interpretations. Copenhagen doesn't make any committment to the existence and nature of both the wavefunction and the collapse process: it says they are just mathematical descriptions useful to predict empirical observations. While Copenhagen interpretation has itself multiple interpretations, it is typically understood as the instrumentalist "shut up and calculate!"

The thought experiment you describe appears to be flawed. According to the principle of deferred measurement, in any quantum experiment you can always assume that measurement (that is, collapse) occours only once at the end of the experiment. Intermediate measurement operations can be replaced by unitary operations and all classical systems involved (automated devices, cats, people, ...) are treated as fully quantum systems whose state can become entangled with the state of the "true" quantum system. This is a mathematical theorem of formal quantum mechanics, hence it holds in all interpretations (at least approximately, see below). You can't use internal measurements to distinguish between interpretations, at least not as trivially as in your proposed experiment.

Objective collapse interpretations like Penrose's predict that closed-system evolution becomes non-linear above a certain scale or in certain conditions, hence they are in principle distinguishable from the other interpretations. Testing would require preparing some specific kind of coherent superpositions of the state of large-scale quantum systems, keeping them significantly insulated from decoherence for a time long enough to make the nonlinearities non-negligible and then measuring. The results should deviate from the predictions of standard quantum mechanics.