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.