According to classical electromagnetic theory, accelerated charges should emit radiation and lose energy. The reason given in my book why atoms don't emit radiation (say, when the atom moves along a circle) is because the atom is neutral. I can understand how this works for a neutral particle like a neutron but the atom has constituent charges within in. How can the "presence" of an opposite charge nearby stop what seems to be an intrinsic process independent of the surroundings? Do the electrons and protons emit radiation that destructively interferes or something of that sort?
Your book is wrong. "Atomic Bremsstrahlung" is a thing, and occurs when an neutral atom has a dipole moment and is accelerated somehow.
As a practical matter, situations in which something as massive as an atom is accelerated up to a sizeable fraction of its rest-mass, while at the same time not being ionized by the forces involved are pretty few and far between, so the phenomena doesn't come up that often.
But there's nothing, quantum or classically, about being part of a larger neutral system that prevents a charged particle from radiating if the whole system is accelerating.
According to classical electromagnetic theory...
I think this is the key assumption you are building your question on, and for atoms/nucleons/electrons/everything smaller this assumption just doesn't hold true. All these objects have to be described with quantum mechanics, so there is no trajectory of a localized charge or something alike - all you're left with is a probability density to find a localized charge at a certain point in space-time. This probability-density might change over time, but there is no classical acceleration, there is not even a particle...
Down to quantum size as atomic scale, particle idea does not work anymore. Instead in quantum mechanics wave function and eigenstate are basic concepts. Physical quantities familiar in classical theory are operators and their expectation or average value is what one observes in the classical sense. To tell if an electron is moving it's the electron density distribution that one should look at.
When you say an atom does not emit EM radiation, you assume it sits in the ground state, the lowest eigenstate. The electron density of an eigenstate is invariant as time flies. Interpreted classically, as you like, the electron in the atom is sitting there doing nothing, so no radiation occurs. On the other hand, atoms do radiate in many situations. That's because electrons jump between eigenstates. Now the electron density is changing with time, meaning classically they are accelerating and so emitting photons.
As for free electrons, such as in an electron beam, their wave function (a wave packet) is not confined. Therefore they don't have discrete eigenstates to stay in. They are traveling all the time, more similar to the classical world. When the velocity (still the expectation value) is not constant, radiation will happen.