If the literature is such a poor state, I'm a little puzzled as to why. 

Maybe for SU(3) it should be clear in principle what is happening? The trace is an invariant of conjugacy classes, and the Weyl integration formula should reduce any issue to an integral over a two-dimensional region. Namely the maximal torus T is acted on by the Weyl group W, and apart from stuff of Haar measure zero the quotient can be represented by an explicit fundamental domain. The push forward of Haar measure on SU(3) has a density with respect to Lebesgue measure on the fundamental domain that was written down by Weyl (if not before). The trace is also a function on this fundamental domain. Points in T can be represented by three angles summing to zero mod 2pi. 

Am I being slow? The generic conjugacy class is not a difficult thing to understand, once having thrown out something of measure zero.