Game is not a topos

Answer

The category $\mathbf{Game}$ is not cartesian closed and therefore is neither an elementary nor a Grothendieck topos. This is Remark B.27 of Games as recursive coalgebras.

The obstruction is failure of a product functor to preserve colimits. For the game $\mathbb S_2$ used in the paper, the quotient of its canonical $\mathbb Z/2\mathbb Z$-action is $\mathbb S_1$, whereas the corresponding quotient after multiplying by $\mathbb S_2$ has six elements and is not $\mathbb S_2\times\mathbb S_1$. Hence $\mathbb S_2\times(-)$ has no right adjoint.

This does not contradict the existence of a subobject classifier or local finite presentability. It also does not contradict closedness for Conway addition: the categorical product and Conway addition are different operations.

Separate question

A suitable completion might be a topos even though $\mathbf{Game}$ itself is not. Note 42 concerns such a completion. Any claim about it must specify the completion and embedding and provide its own proof; it does not reopen the answered question above.