Problem designing a specific filter
so that $y[n]$ in the second diagram (the one on the right) is exactly the same as $x[n]$ in the first diagram (the one on the left).


Considering that $H_{c1}$ is per your very own definition $\ne H_{c2}$ means that without restricting your system's input $r_c$, there's nothing you can do. You will have to implement a filter with transfer function $H_{c1}$ if you need a system with transfer function  $H_{c1}$; there's no way around that.

Now,  $H_{c1}$ being impossible with finitely long filters (as it is infinitely sharp in frequency domain), this is impossible.