Safety of a private key when knowing the full content of an encrypted message in case of fixed initialization vector
What you're describing is called a known-plaintext attack, and any secure encryption algorithm should prevent this. If we take an encryption function with no IV $C = ENC(M, K)$ we should get a system where knowing both $M$ and $C$ reveals nothing about $K$.
We can expand this principal to $C = ENC(M \oplus IV, K)$ and simplify it back to the example without an IV; $M \oplus IV$ simply becomes our new $M$ (you could say $M' = M \oplus K$). This principal applies also for $C = ENC(M \oplus IV) \oplus IV$.
Any encryption algorithm which fails to achieve this property cannot be considered secure, and as such in your scheme $K$ will not be leaked if the encryption algorithm used is secure.
However, using a fixed IV is bad practice since it allows statistical analysis attacks (read about it here). While $K$ may not be leaked $M$ might leak from statistical patterns.