Is the mass defect in Einstein's $E=mc^2$ the mass of the force-carrying particles within the nucleus? Basically, what the title says. Is the difference in mass between the sum of the masses of individual nucleons and the nucleus itself the mass of all the force carrying particles I.e. $W$ and $Z$ bosons?
 A: You are mentioning mass defect, and as it says, it is a defect, meaning the mass of the nucleus is less then the mass of the separate constituents.

Nuclear binding energy is the minimum energy that would be required to disassemble the nucleus of an atom into its component parts.
The mass of an atomic nucleus is less than the sum of the individual masses of the free constituent protons and neutrons
This 'missing mass' is known as the mass defect, and represents the energy that was released when the nucleus was formed.

https://en.wikipedia.org/wiki/Nuclear_binding_energy
As you can see it is related to the nuclear binding energy, that is the residual strong force, that binds the protons and neutrons together to form a nucleus. The force carrying particles you mention are related to another force, the weak force.
The force between the protons and neutrons (residual strong force) is modeled in mathematics with virtual particles. But even if you would like to relate this binding energy to a force carrying particle, these would be the force carriers of the residual strong force.

The nuclear force (or nucleon–nucleon interaction or residual strong force) is a force that acts between the protons and neutrons of atoms.
The nuclear force occurs by the exchange of virtual light mesons, such as the virtual pions, as well as two types of virtual mesons with spin (vector mesons), the rho mesons and the omega mesons.

https://en.wikipedia.org/wiki/Nuclear_force
