Suppose an environment, containing some mass A, is in thermal equilibrium at temp T0 in lab frame S. Rapidly spin-up A to some constant circular orbital speed u about a fixed axis in S - radius can be arbitrarily large so as to avoid any issues related to centripetal acceleration/stresses. This avoids Twin Paradox style ambiguities. Unambiguously, averaged over a per-revolution basis a clock co-moving with A's instantaneous proper frame S' runs slow as seen in S by factor 1/γ = √(1-u2/c2). It also answers the problem of obtaining sufficient time to reestablish thermal equilibrium. In S' A has initially the same proper temp T' = T0 as when formerly at rest in S. In S net thermal energy W of A has been boosted by the factor γ. But A's clock is running in S at a rate the inverse of that same factor. Consequently A is radiating into S an invariant average quantity of thermal power dW'/dt' = γdW/dγt' = dW/dt. Surely an unavoidable conclusion. If the net radiated power is invariant, so surely is A's 'average' temperature T = T' = T0 seen in S[2]. Which conclusion disagrees with both Moller and Planck; sitting between as the geometric mean of those two's contradictory relations.
As has already been pointed out, there is an angular redshift/blueshift spectral spread of environmental radiation seen by A in S'. Clearly if A has a finite aspect ratio (i.e. does not form a circular loop spinning about it's major axis) then the leading edge will heat, trailing edge will cool, and an axial heat flow ensues until a proper equilibrium temp distribution is reached. At least to first order, seems reasonable this new distribution will not alter that T = T' = T0[1]. Prior to equilibrium, A was experiencing a retardation force opposing circular speed u owing to the bias of blueshifted (increased) radiation pressure on it's leading face vs redshifted pressure on the trailing face. After equilibrium, that net force should increase not reduce. Which should also be reflected in the blueshift/redshift angular bias in A's radiant output seen in S.
[1]: But some more thought and this conclusion fails to properly account for the inherent asymmetry owing to circular motion. In S', A must determine that the per-revolution mean clock-rate of thermal oscillators in S has increased, not decreased by factor γ. Implying that as determined in S', radiant input into A from environment has increased by factor γ2. One γ for clock-rate, another for boosted number density of oscillators seen in S'. Hence an initial net inflow over outflow into A by a factor γ2-1. Transformed back into S, this net inflow is reduced by clock-rate factor 1/γ, leaving a net initial inflow factor (γ2-1)/γ. Final result then is a boosted temperature of A seen in S. Think that's now about right. Which sort of complicates things and makes circular motion somewhat special or at least distinct from uniform rectilinear motion case where clock-rate asymmetries do not enter. Then again it can be well argued uniform rectilinear motion is an unreal scenario.]
[2]: Forgot something: Net power flow (initially - neglecting subsequent cumulative effect of radiant imbalance as per [1] above) is invariant, but there is length contraction of A in S by factor γ, so radiating surface area is reduced. Hence power density is boosted, and that directly implies a boosted temperature of A seen in S. However this cannot be a simple γ-factor affair as geometry of A strongly enters - for a given γ factor, change in net surface area of an already squat-shaped A (say a thin disc) is much less affected than say for a slender rod shape.