Is it considered a sin against God to pursue the bakery owner—who is a relative—after rat droppings were found inside his bread?

First: The rahm (kin) are relatives from both the father's and mother's sides, and this includes the daughter of a paternal uncle's daughter. It is obligatory to maintain ties with them.

Second: Allah Almighty has made it obligatory to fulfill trusts and to bear witness with justice, even if it is against one's closest relatives, as stated in His saying: (O you who have believed, be persistently Qawwameen [firmly standing] for Allah, witnesses in justice, even if it be against yourselves or parents and relatives). What your relative's husband did by neglecting the cleanliness of the bread is a reprehensible act that must be denounced and changed, especially if you are in a position of authority, because that falls under helping the oppressor by preventing him from his injustice, and it falls under changing the reprehensible with one's hand for whoever is able, and this is an obligation upon you.