According to the common spelling logic in Russian soft and hard sounds are spelled by the same letter, sounds become soft when their letters are followed by е, ю, я, ё, и, ь. So, we can dispose of щ completely.
Here is the acoustic way of spelling: Ши да каша пишя наша.
It should be "Шши да каша пишшя наша" because щ represents a double soft ш. I vote for щ.