I don't know of any indications that the feminine counterpart of masculine nouns in Biblical Hebrew can be used as a diminutive. I previously suggested that the case of e.g. Modern Hebrew שׂק is not directly related to gender but to personal names (that this only seems to be possible with ־ת would support this, although I must say I do not know Modern Hebrew).
Rather, in Classical Hebrew we see traces of an older system where grammatical gender had some semantic function. In this system, feminine verbal substantives are often associated with collectives or abstract nouns; on the other hand, a feminine ending added to a masculine collective can indicate a particular item of the collection. Thus, in this system, gender interacts with plurality and definiteness (although it is as far as I know not clear exactly how this system worked originally). For a brief discussion about semantics of gender, see Joüon-Muraoka §134n–p.
Seeing that we have a personal pronoun with demonstrative force here (היא), I would suggest that the feminine here indicates the particular food with regards to the general אכל-event that is introduced by the verb (ויאכל). I had a quick look at the instances of אכל and almost all talk about food in general, supporting this. (The occurrence in Gen. 41:36 is translated with a demonstrative in the KJV, but this demonstrative is absent in the Hebrew.)
In all likelihood, had the text used אכל in 1 Kgs. 19:8, this would have been understood the same way. Although it is possible that the feminine was used to exclude some other interpretations, e.g. that the eating and drinking continued during the forty days (excluded, because the feminine indicates a single unit of food and will therefore be consumed in one meal).