You're not off your rocker, you just need to check your math about the speed getting faster than light. The core temperature of the Sun is a little over 10 million Kelvin, which means the "average velocity of a hydrogen nucleus" is about 500 km/s. The speed of light is 300,000 km/s. Now you said "hundreds of millions of Kelvin", which can happen in the cores of massive stars nearing supernova, but even that gives nuclei at only about 1% of the speed of light. The electrons can get relativistic, especially in the cores of evolving stars (and when they make white dwarfs), but they don't have much mass-- the only times the nucleons get relativistic is in neutron stars and black holes. Your numbers about 1/2000 or so of the mass of the Sun being thermal is a bit high too-- it's more like 1 part in a million (and the negative gravitational self-energy reduces the mass more than this increases it.) But still your point does have an important application-- mass does get altered in the way you imagine, and it can sometimes matter: the mass of a neutron star does require relativistic corrections (but it is not more mass than you'd think, it's less, owing to gravitational self-energy reductions in the mass).