The rotational temperature is the temperature at which the thermal population of the rotational states is such as to give rise to the observed rotational spectrum, in terms of the relative intensities of the different transitions. So calculate, in terms of T, the Boltzmann distribution of the populations of the initial rotational states (taking into account degeneracy), and hence the relative intensities of the transitions, particularly the second and third lines.

I agree that your value of ΘR seems quite high; without knowing the molecular constants, I did an order of magnitude estimation and came out with something of the order of 1K, which seems more sensible. Check your maths.