If you have a $G$-bundle over $M$ (without boundary) then this corresponds to the homotopy class of a map $\gamma:M\to BG$. It is known that for $G$ simply connected, any $G$-bundle over $M$ is trivializable. One way to see this is by obstruction theory. The other is to notice that $\pi_i(BG)=\pi_{i-1}(G)$ and for $i=1,2,3$ this is zero. Therefore, one can get a cellular model for $BG$ which has no $3$-cells. Therefore, $\gamma$ is homotopic to a cellular map $\gamma':M\to BG$ which is necessarily constant. Since $\gamma_\ast[M]=0$ in $H_3(BG)$ by Hurewicz and the previous observations, there is a singular $4$-chain $B$ with boundary $\gamma_\ast[M]$. Then using the necessary pullbacks you get what you want.