No, the second term does not need to be radiation pressure, it is pressure of any kind. That includes radiation pressure, but in most cases that is negligible compared to regular old gas pressure. If the temperature is regarded as known (which is the standard way of visualizing the situation, though it leads to some awkward misconceptions that I won't go into here), then we typically use the ideal gas law to get the gas pressure, and that is usually the P in that equation.
By the way, I think I understand where the questioner is coming from when they wonder about "what force is represented by that last term." Which of the fundamental forces is that, gravity or electromagnetic or what? The answer is, it is not any of the fundamental forces, because it is only a "force" in the sense that we have already adopted what is called the "fluid picture". In the fluid picture, we average over a small volume, and get coarse-grained quantities like temperature and pressure and density. But something else happens when we do that averaging-- any process that transports net momentum into our averaging box is going to get interpreted as a force on that box, even though there is no fundamental force there at all. Gas pressure is usually just the momentum that is transported into the box by virtue of the fact that the particles entering the box have more momentum in a given direction (say, outward, as in that equation), than the particles leaving the box. A great but fascinating subtlety arises here when we recognize that the sign of the outward momentum is of great importance-- particles that exit the box downward will remove negative outward momentum, while particles that enter the box upward will bring in positive outward momentum, so the "bottom side" of the box is responsible entirely for increasing the outward momentum in the box, even though there is no net transport of particles. That's an outward force, called "pressure" when you divide by the area of the bottom side of the box. Similarly, the top side of the box will be responsible entirely for decreasing the outward momentum in the box. That's also pressure, when divided by the area of the top of the box, but that points downward, so to get the net force, you need to subtract the two, and that's why it is the gradient in the pressure that gets interpreted as a force in the fluid model of a gas.
Note also that no collisions are required-- the equation holds as soon as you adopt the fluid model. Collisions just help that model hold true, because the model generally assumes the particle distribution function is essentially isotropic in the fluid frame, which gives us the necessary concept of "isotropic pressure", such that P is just a number. This can also cause confusion-- note that pressure is isotropic, but is still responsible for an upward force if you consider the bottom face of the fluid cell, and a downward force if you consider the top face. So you have an isotropic effect that induces forces by virtue of its spatial gradient, and it is not a fundamental force, it is an effective force that appears in the fluid description.
