Does a charged particle in motion have an additional energy associated with it's motion due to this magnetic field?
For extended particles with finite charge density, Poynting theorem is valid inside the particle and thus can be used to derive expression for EM energy. This energy is only partially in the particle, the rest is in the surrounding space.
Since some energy is in the particle as well, it makes sense to say the particle has additional contribution due to EM energy inside it.
Is this (positive) contribution of energy higher when the particle moves? I do not see any easy way to find out, other than directly comparing the two energies.
To find out these two numbers, you may do the following calculation. Calculate Poynting energy in the region occupied by a uniformly charged sphere at rest. Then calculate the same integral (with magnetic contribution) in other frame where the sphere moves with constant velocity. The particle will have shape of oblate ellipsoid and the fields will change according to the Lorentz transformation, so the answer is not immediately obvious and probably will require some calculations.
This path seems demanding but should be manageable, if not analytically then with help of computer. The disadvantage of this approach is that it depends on specific model of charged distribution. Another particle which does not have uniform charge distribution will need another calculation.
For point particles, Poynting theorem is not valid at points where the particles are and thus cannot be used to derive expression for EM energy. Another work-energy theorem can be derived for point particles that have no self-interaction. Theory of charged point particles has been published many times, for example:
J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.
In English, this article also explains it concisely:
R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Let-
ters, 8, 3, (1964), p. 185-187.
There is no self-action and the field of one isolated particle contributes no energy in this kind of theory.