I know what you mean about not being able to hear one's own accent; I guess what I should have asked is whether in your dialect (I don't mean your folk group's speech, but the dialect you grew up in where "incy wincy" was the normal way to sing about a small spider), your native dialect, whether "pit" and "peat" are homophones, i.e. pronounced the same. If so, then "incy wincy" would be a very reasonable way for it to be spelt (or spelled, as we write it over here). Vowels are subject to modifications much akin to the Folk Process.

In American English there are some dialects where "marry merry Mary" is a series of three identically pronounced words, others where each is completely distinct from the others, and still other dialects where two are identical and the third distinct. In my own speech there is an audible difference if I speak very slowly, but at normal conversational speeds the three sound alike.

Liland