Build a moduli space using homotopy theory.

Question

I'd like to build ''with hands'' the moduli space of stable principal $U(1)$-bundles over a Riemann surface $M$ of genus $2$. I have the follow idea: We know that the isomorphirsm classes of principal $G$-bundles on a topological space $M$ are in one-to-one corrispondence with the classes of homotopical maps from $M$ to the classifying space $BG$: $$ \mathcal{P}_{G}(M) \simeq [M,BG] ,$$ so, in our example we have $$ \mathcal{P}_{U(1)}(M) \simeq [M,\mathbb{C}P^{\infty}] .$$ Now if I find a method in order to describe this space as variety I find the moduli space. We have the follow theorem of Thom: Let $\pi_k(Y)=0$ for $k \ne n$ and $\pi_n(Y)=G$, let $X$ be a finite $CW$-complex, then $$ Map(X,Y)=\prod_k K(H^q(X,G);n-k).$$ So, having that $\mathbb{C}P^\infty$ is a $K(\mathbb{Z},2)$ space, we have $$ Map(M,\mathbb{C}P^\infty)=K(\mathbb{Z},2) \times K(\mathbb{Z}^{4},1) \times K(\mathbb{Z},0) .$$ so $\mathcal{M} = Map(M,\mathbb{C}P^\infty)=\mathbb{C}P^\infty \times S^1 \times S^1 \times S^1 \times S^1 \times \mathbb{Z}$. Can I conclude that $\mathcal{M}$ is the moduli space of stable principal $U(1)$-bundles on $M$?