It is well known that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}_{\partial D^n}(D^n)$ of the orthogonal group and the diffeomorphism of the disk $D^n$ with are the identity in a neighborhood of the identity.

Does the classifying space $B\operatorname{Diff}(S^n)$ have the homotopy type of $BX$? If not, do we at least have an isomorphism on the level of cohomology: $H^\ast(B\operatorname{Diff}(S^n);\mathbb{Z})\cong H^\ast(BX;\mathbb{Z})?$

Of course, if there was a homotopy equivalence $\phi:X\rightarrow \operatorname{Diff}(S^n)$ which is simultaneously a group homomorphism, the answer would be "yes". But the only homotopy equivalence I'm aware of (which comes from the link at the start of this post) is not obviously homotopic to a homomorphism.

The linked homotopy equivalence has the property that $\phi$ restricted to either factor is a homomorphism. Using this, it's easy to show that $H^\ast(B\operatorname{Diff}(S^n);\mathbb{Z})$ surjects onto the cohomlogy of each factor of $BX$, but I've been unable to piece this together to get what I want.

Motivation: I'm trying to understand characteristic classes for smooth sphere bundles whose structure group does *not* reduce to $O(n+1)$. It's clear that, except for a few small $n$, one should get "extra" characteristic classes, but I'm trying to understand if one also gets an analogue of Stiefel-Whitney, Euler, and Pontryagin classes satisfying all the "usual" properties, e.g., the mod $2$ reduction of the Euler class should be the top Stiefel-Whitney class; Stiefel-Whitney classes satisfy the Wu-formula, etc. I'd also be happy with pointers to the literature.

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