Is there a thermodynamic heuristic argument on why a redshifted blackbody spectrum is a blackbody at a new temperature?

This is a neat fact! I think the first time it's usually encountered is in a cosmology course, where the expansion of the universe keeps the CMB temperature well-defined. Here's an argument why.

Consider adiabatic expansion of a photon gas at temperature $T$ from the standpoint of kinetic theory. In this point of view, each photon is a particle rapidly bouncing back and forth. Unlike for a classical ideal gas, every photon has exactly the same speed, so every photon must lose the same fraction of its energy. (This is because each one collides with the walls an equal number of times, picking up the same relativistic redshift factor every time.)

Thus adiabatic expansion causes the frequency shift you're talking about. Now we just have to show that adiabatic expansion also keeps the temperature well-defined. But this is immediate by thermodynamics: we can already run a Carnot cycle with a photon gas. If, after each adiabat, the gas were not in thermal equilibrium, we could do additional work by using this temperature difference, contradicting the Second Law.