Look at pretty much any introductory text on QM and the Copenhagen interpretation (or the “orthodox” interpretation) is presented as an objective collapse theory, with collapse being a physical process that takes place upon measurement.

That is perhaps a common misconception in popular science publications aimed at non-technical audiences, but I’m not aware that it’s prevalent in technical literature. Even if it was, that’s not a good reason to further the misuse of terminology.

As for your point 2, it just isn’t true that all collapse interpretations assume that collapse only takes place at the end of the experiment.

It doesn’t matter. All interpretations must agree with the predictions of the theory, at least in all the cases that have been practically testable so far. The experiment you proposed predicts the same results whether or not you shield the intermediate observer from decoherence. If your math predicts different results, then there must be some mistake in it.

Also, if collapse is supposed to take place only at the end of a measurement, how do objective collapse theories make sense of phenomena like the quantum Zeno effect, where measurement is taking place continuously throughout the course of the experiment?

Why wouldn’t it make sense of it?

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.