If we know that the virtual photon emitted in an electron scattering Feynman diagram violates the energy and momentum conservation laws (though temporarily), why do we accept it as a feasible diagram representing another perspective of reality?
Internal lines are not particles: they are contractions of interacting fields (in the Dyson-Wick sense). We call them virtual particles to make it easy and efficient to communicate with each other, but they are not to be thought of as particles in no sense whatsoever. For example, the momentum carried by an internal line is just a Fourier variable, not the eigenvalue of the momentum operator (in this sense, we could write $\xi$ instead of $p$ for internal lines), and therefore it is not to be thought of as a physical momentum.
See this other answer of mine if you want some more details.