This question confuses two frameworks, the classical thermodynamic one with the quantum mechanical one.
A perfectly homogenous radiating body belongs to classical thermodynamics.
Photons belong to quantum mechanics, as well as the uncertainty principle.
Lets look at the problem classically. Electromagnetic radiation is continuous from a classical body and therefore theoretically there is no uncertainty other than the measurement uncertainty given by the method of determining the position of the body, since, as you say, at the same time radiation is leaving equally and oppositely.
The problem quantum mechanically can only be looked at quantum mechanical dimensions, so the body should have nanometer size and smaller . Let us assume a crystal for simplicity, because it has a coherent wave function, and no input energy is keeping it in equilibrium with the environment. The crystal wave function has higher energy levels the higher the temperature, and empty levels below to which it can relax radiating a photon. In these dimensions the quantization of radiation is relevant. There will be a time constant for a photon to be released because the body fell from a higher vibrational level to a lower one, and this will not be compensated at the same delta(t) by an equal and opposite photon in the other direction, because photon emissions are random depending on the wavefunction of the crystal . whose square has the probabilities of emission.
So in the quantum mechanical frame the "opposite" emission of photons cannot hold in a delta(t) so the photon momentum will impose an indeterminacy, assuming the center of the nano crustal to begin at (0,0,0). Using the Heisenberg uncertainty principle, the momentum of the photon is
the uncertainty in momentum is proportional is h*delta(lamda)/lamdda^2 and this will set the uncertainty in position x ,that cannot be known even in theory. This ends up to
delta(lamda)*x> lamda^2 , x>lamda
so the position x cannot be known better than the wavelength of the emitted photon.
This intrinsic quantum mechanical bound of order of nanometers can never be reached in accuracy when measuring the position of a classical body, more so of an astronomical size body. It is irrelevant.