I'm sorry Grishka, I hadn't seen this thread when you posted on it but I can't help. I've known it all my life (♪ I was born, ♫ born in the 50's), it's in some of my songbooks but none mentions a date or a place. I read discussions online about "venir à lui" (come to him) vs "venir à l'huis" (come to the door, but "huis" is really outdated!) and some other people did too (Cf the comments in our MLW page) Someone sent us a second verse for it...

Quand ils furent en tête à tête
Le cerf lui fit grande fête
Et le chasseur tout marri
Retourna chez lui
"Cerf, cerf, grand merci,
Le chasseur est loin d'ici!
- Lapin, lapin, réjouis-toi
Et reste chez moi."

in which "marri" is also an outdated word (was in disuse in the 1900's). Now... the question is "Is the song old and were these two old words used normally or did someone make it sound older than it really was by using outdated word?" About whether it was French in the first place, I have no idea.