[lcnPylGDnU4H9OF]

It sounds like reduplication is about individual words being repeated rather than a phrase.

> In linguistics, reduplication is a morphological process in which the root or stem of a word, part of that or the whole word is repeated exactly or with a slight change.

The ablaut section also seems to suggest that "eat" should come before "have" anyway.

> In ablaut reduplications, the first vowel is almost always a high vowel or front vowel (typically ɪ as in hit) and the reduplicated vowel is a low vowel or back vowel (typically æ as in cat or ɒ as in top).

I suspect you feel that it flows more smoothly because it's more familiar. You have to stop a brain process that's become a bit automatic to say that phrase and instead say it slightly differently.