Two possible ideas. One is that you can realize $K(\mathbb{Z},3)$ as the quotient $U(HS)/PU(\infty)$ where $U(HS)$ is the unitary group on the Hilbert space of Hilbert-Schmidt operators. However, I don’t know if you can see the group structure in this model to make a principal bundle. This construction is in https://arxiv.org/pdf/hep-th/9702147.pdf. 

There is also Andre Henriques’s conjecture that the outer automorphisms of a hyperfinite type III factor model $K(\mathbb{Z},3)$. See https://mathoverflow.net/questions/44045/naturally-occuring-k-pi-n-spaces-for-n-geq-2