As the question states, does salt dissolve in fat when heated. If no how can we get it to dissolve in fat and distribute through the fat like it does with water?
Heating will only change the viscosity of the fat, but it has little to do with solubility in this case. Solubility is not a function of viscosity, but different interactions including solvation, which wouldn't take place in fat as much as it would in water. I'll tell you more: acetone is less dense, less viscous, and even polar, but NaCl is still only slightly soluble in it.
To do what you ask, about distribution, I would probably dissolve salt in something it is well soluble, and add some surfactants. Fat and water (?) would make a homogenous mixture, and with right choosing of ratios of fat/water/salt, you'll get almost what you want.