How do we know relativistic mass also includes potential energy? From Special Relativity it is shown that relativistic mass increases with kinetic energy, but how do we know potential energy is also part of relativistic mass? Is this a derived conclusion, or should it be treated as a first principle whose correctness is only verified by experiments (for example the measured mass of composite particles)?
 A: 
How do we know mass also includes potential energy?

Mass is connected to potential energy,  

From Special Relativity it is shown that mass increases with kinetic energy,

This is the relativistic mass that increases with with kinetic energy, i.e. the mass equivalent in Newton's $F=ma$

but how do we know potential energy is also part of mass?

It is not. In Newtonian physics which macroscopic bodies obey,  mass is an additive quantity, and is accurate to very great accuracy, that is whyArchimedes principle works.  One has to go to nuclear and elementary particle sizes to find a connection between mass , energy and potential energy, in the realm of quantum mechanics.
Mass at this level is an invariant to the Lorentz transformations , the length of the four vector which describes a system, and there potential energy enters in the calculations. 
A hydrogen atom is lighter in mass than an (electron +  proton), because the electron falling to the ground state of the potential well releases a photon, and the photon four vector takes away energy, which is missing in the summation of the four vectors of the bound electrons and protons, making the atom with smaller invariant mass than the sum of its parts.

(for example the measured mass of composite particles)?

take the deuteron, so as to have some numbers in kg,The free proton and neutron masses  add up to $3.3474 * 10^{-27}$kg, the mass of the deuteron is $3.3435 * 10^{-27}$kg .
It is thinking in terms of addition of the four vectors that will give the correct algebra of special relativity and the relation of masses to binding energy/potentials.
Edit after comment:
As Safesphere's comment indicates and this link studies, the case of general relativity, the gravitational potential and mass energy becomes more complicated.
The Newtonian case is here, again by Safesphere.
He gives a good precis in his comment:

this is gravitational binding energy that creates a mass defect similar to your electromagnetic example. The difference is that the electromagnetic field is created by charges, not masses, so the mass defect cannot be absorbed by the field, but must be emitted as a photon. In contrast, gravity is created by masses, so the mass defect of the binding energy is simply absorbed by the field itself. .. The effect in GR is only stronger

