Can we calculate the average temperature of the heliosphere?
The heliosphere is mainly defined by the region dominated by solar wind against the interstellar medium.

"The solar wind is divided into two components, respectively termed the slow solar wind and the fast solar wind. The slow solar wind has a velocity of about 400 km/s, a temperature of $1.4–1.6×10^6 K$ and a composition that is a close match to the corona. By contrast, the fast solar wind has a typical velocity of 750 km/s, a temperature of $8×10^5 K$ and it nearly matches the composition of the Sun's photosphere."

Hence with 1 million Kelvin we get a good estimate of the average temperature of the heliosphere.

The temperature is the freeze-in temperature derived from carbon ion charge states measured by SOHO. Hence it's not the black-body temperature, but a reasonable definition of temperature for solar wind.

"Coronal temperatures are inferred from density ratios of adjacent charge states. The freeze-in temperature derived from a given density ratio is the electron temperature that reproduces this ratio in a static situation. For the analyzed iron charge states we use the ionization and recombination rates of Arnaud and Raymond [1992]". More details about SOHO's temperature measurement here.

If the heliosphere would be in the thermodynamic equilibrium, according to Planck's law or Wien's law, you would get the peak at about 3nm, in the soft x-ray range. But the equilibrium assumption probably doesn't hold.

For the hot interstellar medium extreme ultraviolet radiation measurements have been tried (CHIPSat), some successful (EUVE).
Hence, in an ideal environment we would measure the radiation intensity across the electromagnetic spectrum, look for the peak intensity, apply Wien's law, and retrieve the  effective temperature, which would be identical to the black body temperature in an ideal world.

The plasma of the heliosphere is very rare, therefore almost transparent. That's why we don't feel that heat of about 1 milion degrees. Microwaves play a minor role in that temparature range.

A temperature profile of Sun's corona and heliosphere is shown e.g. on pages 42 and 47 of this survey.