Continuing my previous question, given a compact, connected Lie group $G$, there is a sequence of maps $$R(G) \to \hat{R}(G) \overset\sim\to K^*(BG) \to \hat{H}^*(BG;\mathbb Q)$$ apparently first discussed by Atiyah and Hirzebruch. The induced map $K^*(BG) \otimes \mathbb Q \to \hat{H}^*(BG;\mathbb Q)$ isn't an isomorphism; the basic point is that $\mathbb Q[[u]]$ is much bigger than $\mathbb Z[[u]] \otimes \mathbb Q$.
The obvious next question is what happens if one replaces $R(G)$ by $R(G) \otimes \mathbb Q$ at the beginning. I suspect (but am not sure) that things "work" then. To wit, approximate $EG \to BG$ by a sequence $E_n G \to B_n G$ of compact $n$-universal principal $G$-bundles (i.e., such that $\pi_{\leq n}(E_n G) = 0$). The sequence $$R(G) \otimes \mathbb Q \to K^*(B_n G) \otimes \mathbb Q \overset\sim\longrightarrow H^*(B_n G;\mathbb Q)$$ then should induce in the limit isomorphisms $$(R(G) \otimes\mathbb Q)^\wedge \overset\sim\longrightarrow \varprojlim \big(K^*(B_n G) \otimes \mathbb Q\big) \overset\sim\longrightarrow \hat{H}^*(BG;\mathbb Q).$$
1. Is this right?
If that is correct, there's something else going on I don't understand. Because $\hat R(T)$ is a integral domain and torsion-free over $\mathbb Z$, the same holds of the Weyl invariants $\hat R(T)^W \cong \hat R(G)$. This should mean the obvious map to $\big(R(G) \otimes \mathbb Q\big)^\wedge \cong \hat H^*(BG;\mathbb Q)$ is an injection. This is all very well and satisfying if $\pi_1(G)$ is torsion-free, for then it is known that $R(G)$ is a polynomial ring on $\mathrm{rank}(G)$ generators and so is $H^*(BG;\mathbb Q)$. But this does not hold in general; for example, it is known that $$R\big(\mathrm{PSU}(3)\big) \cong \mathbb Z[X,Y,Z]/(Z^3-XY)$$ although $\mathrm{PSU}(3)$ has rank $2$, and $$H^*\big(B\mathrm{PSU}(3);\mathbb Q\big) \cong \mathbb Q[c_2,c_3], \qquad |c_j| = 2j.$$ So if the claim in my first question is right, the completion process makes one generator and one relation redundant.
2. In this example, what does the completion process do?
I should specify that although I found the description of $R\big(\mathrm{PSU}(3)\big)$ in the paper of Brylinski and Zhang, I'm not clear what relation these generators $X,Y,Z$ have to the fundamental representations and so in particular still don't know what the augmentation ideal is. Is there some place these computations are carried out so I could see what's happening?