The earliest source I was able to find for this particular palindrome is a work by the famous kurdistan author, Shimon Avidani (1881 - 1981), called ma'aseh hadgdolim (parshas toldos no. 13, p. 382). The author brings this from an ancient anonymous text:



> וגם ראיתי כתיבת יד עתיק כך: **רעבתן שבדבש נתבער ונשרף.** גם זה אתה יכול
> לקרוא ישר והפוך, וגם בראשי תיבות ישר והפוך בכל צד ואופן.

The author brings this along many other palindromes by the Ibn ezra, and this may have caused the incorrect attribution of this particular palindrome to the Ibn ezra; although he was undoubtedly a genius it doesn't seem to be his creation.