Alright. Must've made a math error in my rough calculations. Oops! In fact I'm almost certain I know what it was, I probably forgot to transfer between meters and kilometers per second.

Still - if kinetic energy can be considered mass, I have a hard time seeing any large problem in my mass fraction calculations.

I didn't bother calculating the values in the sun, I figured that'd be very small. I took an extreme example, a star with 60 times the mass of the sun and a million times as much energy being produced per second and emitted per second. Depending on how long it takes energy to diffuse out away from the core it would presumably contain at least a million times as much thermal energy stored in the thermal motion of its atoms/molecules/plasma.

Applying the relativistic mass correction to an object traveling at one percent the speed of light I get that its mass increases to 1.00005 times it original value - thats one part in 20,000. In principle it seems to me that the very largest stars with tremendous luminosities and very high temperatures might have mass fractions of thermal energy higher than 1 in 100,000. I know theres pretty much no practical application to this but its something awesome to think about.

Luminosity (J/s) * path time (~100,000 years) = internal heat energy of the star when it is in equilibrium. I still get the same value for the thermal energy of the enormous star in Joules... and thus the same mass increase. I'm not even considering relativistic speed, I'm just considering good old E=MC^2. That IS applicable here, yes?