This is a hoax.
I thought it was a hoax, but having seen the brief text, I now believe that Bambi did send it to Chirac, although not because he thought Chirac was still president. He’s just saying he wants to work with Chirac in the cause of peace, yadda, yadda, and completely ignore the current president. Obama did this to insult Sarkozy, who is (by French standards) a conservative.