Iron is supposed to glow blue-white at around 1600 degrees C. page 7.
Would carbon have that color at that temperature?
Spectral lines of iron
Spectral ines of carbon
Maybe not. Maybe those bright yellow lines would add too much. Carbon might have to be hotter.
Do we have examples of carbon heated very hot? How about a carbon arc lamp? Pass a whole lot of electricity between two pieces of carbon. Some carbon vaporizes, and the hot carbon vapor gets very bright.
Does this look bluish-white to you? Maybe it's possible, and maybe the charcoal has to be vaporized to do it. Can you find a way to get charcoal to do that by burning it with pure oxygen? I don't know. Can you get that color with solid charcoal, or would the carbon vaporize first? Carbon's sublimation point is 3642 C, so maybe.
Sorry I can't give a more definite answer.
(Here's a tip if you want to look for yourself. If you do an image search that includes "white-hot" or "blue-white-hot", be sure to set your censor. For reasons I didn't think of ahead of time, these searches produce a lot of porn.)