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You can construct the map $K(\mathbb{Z},3) \to tmf$ as follows: first there is the String orientation of tmf, which you already mention. This is a map $$ MString \to tmf$$ Then String is by definition a $K(\mathbb{Z},2)$-fibration over Spin. This yields in particular a map $$ K(\mathbb{Z},3) \to MString $$ Then you can construct the map $K(\mathbb{Z},3) \to tmf$ as the composition of the above two maps. In order to extend this constuction you had to find a map $BBU_\otimes \to MString$. I think such a map does not exist, but I am not entirely sure.