If St. Peter's was built with indulgence money, why do Catholics believe in indulgences?

First, admit that there is a real possibility corrupt men in the Church misused indulgence money, because we have had scoundrels in every pontificate; the New Testament itself warns, in Acts 20, that wolves would come in. So even if St. Peter's was built partly with indulgence money by a few wolves in the 1500s, that would be no surprise and does not condemn the doctrine. The doctrine itself was created in the most sincere way: as with giving alms to feed or clothe someone, a heartfelt gift to the Church, which you believe is God's instrument for saving souls through the gospel, is a real sacrifice of your goods, and in return God gives you a reward, not a payment, helping you deal with the temporal punishment due for your sins. There is room for abuse, which is why you must carefully test the people you entrust with Church money, but the abuse of a thing by scoundrels does not make the thing itself wrong. You have to make the proper distinctions.