Cross sections for ($p,t$) two-neutron transfer reactions are calculated in the one-step zero-range distorted-wave Born approximation for the tin isotopes $^{124}$Sn and $^{136}$Sn and for incident proton energies from 15 to 35 MeV. Microscopic quasiparticle random-phase approximation form factors are provided for the reaction calculation and phenomenological optical potentials are used in both the entrance and the exit channels. Three different surface/volume mixings of a zero-range density-dependent pairing interaction are employed in the microscopic calculations and the sensitivity of the cross sections to the different mixings is analyzed. Since absolute cross sections cannot be obtained within our model, we compare the positions of the diffraction minima and the shapes of the angular distributions. No differences are found in the position of the diffraction minima for the reaction $^{124}$Sn($p,t$)$^{122}$Sn. On the other side, the angular distributions obtained for the reaction $^{136}$Sn($p,t$)$^{134}$Sn with surface and mixed interactions differ at large angles for some values of the incident proton energy. For this reaction, we compare the ratios of the cross sections associated to the ground state and the first excited state transitions. Differences among the three different theoretical predictions are found and they are more important at the incident proton energy of 15 MeV. As a conclusion, we indicate ($p,t$) two-neutron transfer reactions with very neutron-rich Sn isotopes and at proton energies around 15 MeV as good experimental cases where the surface/volume mixing of the pairing interaction may be probed.