As you said, you want that your wave function is normalized.
So, it is not correct that the solution outside the wall is $$\psi=Cexp(\mu x)+Dexp(-\mu x).$$ In fact this function diverges for $x\rightarrow\pm\infty.$
In order to keep your wave function normalized, you must use one solution for $x<-a$ and another one for $x>a.$
For $x<-a$, you have
$$\psi_-=Cexp(\mu x).$$
While for $x>a$ you get
$$\psi_+=Dexp(-\mu x).$$
Your wavefunction is thus:
$$\begin{cases} x<-a & \psi(x)=Cexp(\mu x) \\ -a<x<a & \psi(x)=Acos(\lambda x)+Bsin(\lambda x) \\ x>a& \psi(x)=Dexp(-\mu x) \end{cases}$$ In this way you imposed the normalizability request and you can just ask continuity and derivability in $x=\pm a.$