Kinetic energy need not be carried by any special carrier particle. Kinetic energy is inherent in any moving body or particle.
If you look closely at the conservation of energy/work energy theorem, all you get is that $\Delta(\frac12mv^2)=\int \vec F\cdot d\vec x$
The left hand side is (change in) "kinetic energy". The right hand side is change is potential energy. Calling both of these terms as energy is a convenience that makes it cleaner to state the conservation of energy in terms of one quantity ($E$).
Fundamentally, kinetic energy is inherent in any moving body. On the other hand, forces are caused by exchange of gauge bosons, so potential energy is in the end "mediated" by various particles (like the photon). I wouldn't go so afar as to say that it is "carried" by particles, though. The forces are mediated by the particles; but the potential energy is still a property of the system.
Springs work on the basis of electromagnetic energy. (All forces like the stress reaction force, forces between materials when you push two together, etc come from electrostatic repulsion between electrons in atoms.)
Photons carrying heat energy and photons mediating the EM force are very, very different.
When you transfer heat, the kinetic(vibrational) energy of the body is manifested in the form of an emitted photon. All that energy is now the kinetic energy of the photon. The photon finally bumps into some other material and is absorbed, turning into electrostatic or kinetic energy. All photons here are real photons.
On the other hand, the electromagnetic force (and the other fundamental forces) is generally mediated by virtual particles (photons). These live on borrowed energy, and don't contribute to the potential energy of an interaction. The energy of the interaction is part of the system, not really tied to any individual entity.