But it seems to me that the EM momentum it radiates in each direction is balanced by an equal amount of EM momentum radiated in the opposite direction. Is that true?
It is true in the frame where the particle is at rest when it produces the retarded radiation. In other frames, particle moves and its angular pattern of Poynting intensity of radiation is deflected in the direction given by the velocity.
Therefore how can we account for the radiation reaction force using Newton's 3rd law?
Newton's 3rd law does not apply in electrodynamics. Thus we cannot.
P.S. The standard way ...
This standard argument seems to me a bit of an indirect method that does not give a lot of physical understanding.
Can another classical EM argument give a more intuitive picture involving conservation of momentum or do we have to resort to quantum field theory?
The "derivation" of the LAD term via the periodic motion is a very misconceived procedure. It derives one possible expression for the self-force that is necessary to make the model consistent with the Poynting theorem. But when this force is put into the equation of motion, the equation leads to unphysical behaviour.
This is because the LAD force is only approximate expression for the self-force on the charged sphere. There is infinity of other terms that are neglected.
The LAD force was derived with convincing basis originally by Lorentz. He calculated approximately forces of parts of charged sphere on each other and summed these forces. In contrast to non-relativistic mechanics, the EM interaction leads to modification of effective inertial mass, a force term that is proportional to $d\mathbf a/dt$ and to infinity of other terms with higher derivatives of velocity.
This derivation of self-force for charged sphere is well-based in EM theory, but is only approximate since the calculation is very hard and cannot be done exactly without explicit model of the sphere (in relativity, extended charged object cannot be rigid). So far such tractable model of charged fluid forming a compact body is lacking.