Oh wow!

Now I don't know how to derive this exactly, but had a shot anyway. I get that the velocity of nuclei near the centre of the sun are about 400 km/s (which I am roughly calculating based on what I think the velocity relationship is to temperature and atomic mass) which makes their relativistic mass increase only about 2 parts in a million. That is only about 20 times my estimate of radiation and I suppose 200 times yours. OK that sounds about right. But that doesn't allow for electrons which I suppose are doing more like 18,000 km/s which increases their mass by something of the order of 0.3% but they are only 1/1836 of the mass so that makes roughly the same amount again? Total nearer 3 to 4 millionths. Of course the nuclei and electrons away from the centre are going slower, so correct answer maybe still 2 millionths.