Question: "Why was God so irate with David for taking a census of Israel? Does it have anything to do with the law in Exodus 30:12 about the census offering?"
The passage in 2nd Samuel (2nd Kings in the LXX) 24, actually says that God was angry with the people of Israel: "Again the anger of the Lord was aroused against Israel, and He moved David against them to say, "Go, number Israel and Judah"" (2nd Samuel 24:1). Interestingly, the parallel passage in 1st Chronicles 21:1 says "Now Satan stood up against Israel, and moved David to number Israel." Now, this is not a contradiction. We see from the book of Job, 1st Samuel 16:14, and from the Gospels that even the demons are subject to God, and can only do what God permits them to do. But the Scriptures do not specify why God was angry with the people of Israel, though given that it says that "again the anger of the Lord was aroused against Israel", it is likely that the Israelites were engaged in the same kinds of sins (idolatry, laxity, etc) that we see throughout the historical books of the Old Testament.
The commentary from the Fathers that I have available to me does not go into great detail, beyond simply stating that King David was lifted up in pride. Probably he wish to have a census to glory in the strength of the numbers of his people. But the history of the people of Israel should have taught him the lesson that his dear friend Jonathan had learned: "...nothing restrains the Lord from saving by many or by few” (1 Samuel 14:6).
Neither the Scriptures themselves, nor the Fathers (as best as I can tell by what I have available) connect God's anger at David's census with what is commanded in Exodus 30:12-16, but Josephus does:
"NOW king David was desirous to know how many ten thousands there were of the people, but forgot the commands of Moses, who told them beforehand, that if the multitude were numbered, they should pay half a shekel to God for every head. Accordingly the king commanded Joab, the captain of his host, to go and number the whole multitude; but when he said there was no necessity for such a numeration, he was not persuaded [to countermand it], but he enjoined him to make no delay, but to go about the numbering of the Hebrews immediately" (Antiquities of the Jews 7:13:1).
You can read the rest of the story from 2nd Samuel 24 by clicking here; and you can read the parallel account in 1st Chronicles 21 by clicking here. Interestingly, the plague that God sent on the people of Israel as punishment for their sins led to the purchase of the ground upon which the Temple of Solomon was later built.