Not too helpful, but maybe it's a start...
I recall, but cannot document, that some hard-driven bullets like the .220 Swift are nothing but little gilding metal bottles full of molten lead as they zoom downrange. When overdriven and over-spun, they may fly apart, leaving a little trail of blue smoke... which is (again, I am told) a spray of this molten lead in midair.
This "blue trail," without an accompanying hole in the target, has been documented many times in this and other gun sites.
So, in the spirit of helpfulness, I looked it up in Hatcher's Notebook and found he has a rather muddy description of the distribution of the energy available from a pound of powder. Unfortunately, he does not give a direct figure for the temperature of the bullet, only "heat energy due to friction" of 212.0 Calories out of the total of (I think) 2864.0 Calories available in the .50 BMG from the powder charge.
It is hard to tell from his accompanying table of heat distribution in that single .50 BMG round whether this "frictional" heat goes entirely to the bullet or part to the bullet and part to the barrel. If anyone smarter than I can make better sense out of that table, please say so. (Cf. p 399, op cit.) I am sure that if we knew for sure what portion of this 212 Calories actually went to the bullet, calculation of its temperature would not be dificult, knowing its mass and composition.
For those of us who have ever done "The Hot Brass Dance," where an ejected case falls down your shirt, 131 Calories goes into heating the brass case during firing --or about 5% of the total. (Again, for the BMG. I see no reason the proportion itself would not hold up for most CF rifle rounds.)
(Note: Calories with a capital "C" are "kilocalories," that is, 1000 times the heat energy of the "calories" with a small "c" which we use to measure the energy in foodstuffs. I think nowadays they use "kC" for this same "kilocalories.")