Probably incorrect, or at best partial, answer
I suspect that the ending we're looking for might be
DEPART
on the following grounds.
First, of course, it fits the metre and rhyme scheme. Second, it almost makes sense in the context of the verse (though surely at this point the cat has already departed, and is now doing something else entirely). Third, and more to the point, the letters of DEPART can be found, in order, among the initial letters of the lines. One per rhyming couplet, though there's no very obvious pattern to which line is selected from each couplet.
But I'm uneasy because
"pick one initial letter from each couplet" seems too vague a specification -- if this is anything like the right answer, it seems like there ought to be some indication of which letter to pick each time. The first five letters are from the five lines that are almost but not quite lines from familiar children's rhymes -- but the last one isn't, so far as I know.
And also because
as I mentioned above, it doesn't quite work in terms of the story being told. The cat has already departed, and is about to curl up and have a rest. (But maybe RDFozz's suggestion in comments solves this: the cat was about to do just that, but another bigger, more streetwise, one has come along and scared it away.)
So, how else could those letters be obtained?
In the first few couplets the letter selected is the first letter of the longest repeated sequence of letters in the couplet. DIDDLED, EENYM, PUSSYCAT. Unfortunately this breaks down after that; e.g., in the fifth couplet ACE (in FACE and GRACE) is longer than any repeated sequence beginning with R.
In most couplets it's arguable that the scansion is worse (or at least a worse match for the thing being kinda-quoted, where there is one) for the line whose first letter we are taking. "Diddle diddle dumpling" is meant to be followed by "my son John" and instead we have "my silly owner John", three extra syllables. "Eeny meeny miny moe" is meant to be followed by "catch a [disyllable] by the toe" and instead we have "where shall I bite", three syllables too few. "Pussy cat, pussy cat" is meant to be followed by "where have you been?" and instead we have "get away from my feet", two syllables too many. But this rather breaks down in the fifth couplet: "much like John's face" has the same rhythm as "violets are blue".
None of that seems very convincing. Other remarks that haven't helped me at all:
The last couplet has no nursery rhyme reference at all, unless you count the brief reference to Drury Lane (where the Muffin Man lives). The emphasis on feet might suggest something clever involving "feet" in the metrical sense -- iambs and dactyls and all that -- but if so, I'm not seeing it. In a few cases (but not all) there's some approximation to the relevant letter name inside the couplet (e.g., ee*ny, roses **are red). In a few cases (but not all) there's some tangential connection with the relevant word from the NATO phonetic alphabet (e.g., Romeo going along with those romantic roses, the echo of "eeny meeny"). The letters we need are not always the most frequently occurring in their couplets, either throughout or when looking only at initial letters of words.
Another possibility, astutely pointed out in comments by MOehm:
Take the letters not from the rhyme here but from the names of the nursery rhymes alluded to. "Diddle diddle...", "Eeny meeny...", "Pussy cat...", "A tisket...", "Roses are red...", "The muffin man".
