I try to give an answer, welcome corrections!

When we write a state, we must notice the reference frame which the state lies in, because the form of state is diffrent in diffrent frame. Now I give 3 frames $O,O', O''$. Their correlations are $t'=t-\tau'$, $t''=t-\tau''$, with $\tau'=-\infty$, $\tau''=+\infty$. Let's specify that the collision happens in $t=0$, in $O$ frame's view.

When we say $\Psi^\pm$ are tranform as free particles, we actually mean $\Psi^+$ transforms as free particles just in frame $O'$, and $\Psi^-$ transforms as free particles just in frame $O''$.

Turn now to the inner product between in and out states $(\Psi^-, \Psi^+)$. Noticing that the product must be calculated in the same frame, we specify the frame is $O$. Now, we do a transformation $T$, and to see how the inner product changes. Please note, in frame $O$, neither of $\Psi^\pm$ transforms as free particles. However, we can use time translation to take $\Psi^+$ to frame $O'$, and act on it with $U_0(T)$, then take back to frame $O$.

So, under transformation $T$, $(\Psi^-, \Psi^+)$ changes into

$$(exp(+iH\infty)U_0(T)exp(-iH\infty)\Psi^-, exp(-iH\infty)U_0(T)exp(+iH\infty)\Psi^+)$$

Obviously, they are not equal, unless there are some restrictions on $H$.