As far as I can tell (being a non-physicist), the Transactional Interpretation shares the mathematical simplicity of MWI. And here Kastner and Cramer argue that TI can derive the Born probabilities naturally, whereas MWI is said to need a detour through "the application of social philosophy and decision theory to subjectively defined ‘rational’ observers". So maybe TI is simpler.
The "possibilities" they posit seem quite parallel (pardon the pun) to the multiple worlds or bifurcated observers of MWI, so I don't see the philosophical ad...
I think the short version is that you don't need math that covers the wavefunction collapse, because you don't need the wave function to collapse.
For a longer version, you'd need someone who knows more QM than I do.
In non-relativistic MWI, the evolution of the quantum state is fully described by the Schrodinger equation. In most other interpretations, you need the Schrodinger equation plus some extra element. In Bohmian mechanics the extra element is the guidance equation, in GRW the extra element is a stochastic Gaussian "hit".
In Copenhagen, the extra element is ostensibly the discontinuous wavefunction collapse process upon measurement, but to describe this as complicating the math (rather than the conceptual structure of the theory) is a bit misleading. Whether you're working with Copenhagen or with MWI, you're going to end up using pretty much the same math for making predictions. Although, technically MWI only relies on the Schrodinger equation, if you want to make useful predictions about your branch of the wave function, you're going to have to treat the wave function as if it has collapsed (from a mathematical point of view). So the math isn't simpler than Copenhagen in any practical sense, but it is true that from a purely theoretical point of view, MWI posits a simpler mathematical structure than Copenhagen.
The thing that's always bugged me about the MWI is that it doesn't seem physically sensible. If something isn't physically sensible, than you need to check on your model. This happens all the time in physics - there are so many basic problems where you discard solutions or throw out different terms because they don't make sense. This is the path to successful understanding, rather than stubbornly sticking to your model and insisting that it must be correct.
The impression I get is that, if the math leads you to make a conclusion which seems like physical n...
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.
It is true that the historical Copenhagen interpretation -- the one developed by Bohr -- is instrumentalist. But that's no longer what people mean when they refer to the Copenhagen interpretation. 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.
As for your point 2, it just isn't true that all collapse interpretations assume that collapse only takes place at the ...
http://www.scottaaronson.com/blog/?p=1103
Eliezer's gung-ho attitude about the realism of the Many Worlds Interpretation always rubbed me the wrong way, especially in the podcast between both him and Scott (around 8:43 in http://bloggingheads.tv/videos/2220). I've seen a similar sentiment expressed before about the MWI sequences. And I say that still believing it to be the most seemingly correct of the available interpretations.
I feel Scott's post does an excellent job grounding it as a possibly correct, and in-principle falsifiable interpretation.