For a given black body radiation curve, would the changes to the spectrum resulting from cosmological expansion and those from Doppler effects be distinguishable on the basis of the shapes of the resulting curves alone? Or, put another way, starting from the same spectrum, can both processes produce the same observation (for suitably chosen magnitudes of expansion or velocity)?
The redshift due to cosmological expansion is identical to a Doppler shift in its effect on the spectrum of any source. To be specific, both phenomena "stretch" all wavelengths by the same factor.
There's a very good reason for this: in a suitable coordinate system, the cosmological redshift is a Doppler shift.
You'll find statements in some textbooks saying that this isn't true, but a weak version of this statement, which is nonetheless strong enough to explain why the effects on the spectra are identical, is uncontroversially true. To be specific, the redshift of a distant galaxy can be thought of as the accumulation of many infinitesimal Doppler shifts along the line of sight. (Each member of a family of comoving observers is in motion relative to her neighbor, and each can "watch" the redshift build up gradually due to these relative velocities.)
One perspective on this subject (mine, to be precise) can be found in this paper. Even if you don't like our point of view in this paper, our description of and references to other treatments may be of interest.
This addresses the comment to Professor Bunn's answer more than the original question:
Any spacetime described by the FLRW metric, where objects are static in comoving coordinates and redshift is due to expansion of space, can in principle be also described, via an appropriate change of coordinates, by a spherically-symmetric (SS) metric, where objects move along radial timelike geodesics and redshift is due to positional (gtt) and Doppler factors. Since both metrics describe the same physical system, they are observationally equivalent in all respects.
To note, the equivalent SS metric is static, meaning that gtt and grr do not depend on time, only in the empty (Milne) and lambda-vacuum (de Sitter) cases. Moreover, even in the next simplest case, the flat matter-only FLRW model known as Einstein-de Sitter model, gtt and grr cannot be expressed algebraically in terms of non-comoving time, as I mentioned in my answer to this question. I take it as a strong hint that the FLRW model with expansion of space is the real thing, but even if you take a purely utilitarian position, you can at least say that expansion of space is the only conceptual framework with which it is possible to work.