Different kinds of experiments have already been done to verify it and are being done every day. Positron emission tomography measures that gamma ray energies equal to that predicted by E=mc2 are released when electrons and positrons annihilate each other. Even as far back as 1932, the Cockcroft-Walton generator verified E=mc2 to an accuracy of 0.5% by transmuting lithium into helium and measuring the mass and kinetic energy of the resulting particles. E=mc2 is tested every time a nuclear reactor is powered up. We have more than enough information to know that E=mc2 is highly accurate.We don't know if E=mc2 is highly accurate at predicting the effect of T on W.
We don't have data on the temperature measurement of a thermometer in the Sun. All we have are predictions.If you could place a thermometer in the sun, would you ?