We don't break out the energy stored in rotational modes as a separate term when doing an energy balance
Q = mc∆T
If temperature is only the translational motion, you'd need an additional term to account for the energy in the other degrees of freedom, but you don't do this, or need to, because you already accounted for it in the heat capacity.
Equipartition of energy means that you can't separate the translational from the rotational modes in terms of energy. If the temperature changes, all of the modes gain or lose energy, and the temperature is proportional to this.
The distinction shows up in calculations of the internal energy U = aNkT = aPV
Seems to me there's no unambiguous argument in either direction, because the equipartition force the energy to be shared between the modes. I see nothing in literature making the distinction that it's only translational KE in systems with additional degrees of freedom. Nothing shows up in the equations, and IMO it's confusing to make that distinction when it doesn't show up in or matter to the calculation.