For example an electron radiates when accelerated. So does a positron. But is the radiation emitted by accelerated positronium the sum of the radiation emitted by each separately? If not, why not? If so, does this provide a way of testing whether a given neutral particle is composite? For example, does a neutron bremsstrahlung when decelerated?
In the view of classical electrodynamics, electron-positron pair would definitely emit radiation when accelerated. But in quantum mechanics, positronium at the singlet state (spin anti-parallel) with 0 orbital angular momentum will be perfectly spherically symmetric in its rest frame. In other words, it will have no net charge, net dipole moment, net quadruple moment, .... So it will not radiate. Triplet state and excited states may, however.
Neutron will surely emit radiation when accelerated, although very small, because it has non-zero magnetic dipole moment. That is not called bremsstrahlung however.
And this would not be a good way to probe whether a particle is composite or not. The higher-order radiation is small and hard to detect, and it is much more straightforward to simply measure dipole or quadruple moments.