How to decrypt unusual Many Time Pad
This cannot be used to compromise $P_0$. Suppose that, instead of generating $C_0$ and $C_1$ as $R_0\oplus R_1$ and $R_2\oplus R_1$, the attacker instead picked $C_0$ and $C_1$ at random from the set of bit strings the length of $C_2$. This situation is indistinguishable from yours; in both cases, $R_0$ and $R_2$ are independent and follow a uniform probability distribution, and the same is true for $C_0$ and $C_1$. If you can compromise $P_0$ in your case, you can compromise it if the sender had actually generated $C_0$ and $C_1$. But that would mean that sending two completely random strings alongside an OTP message somehow helps compromise the message. Obviously, that's not true; the attacker could otherwise create his own random strings and break the OTP that way.