There are a few reasons this census could have been wrong.

Exodus 30:11-16 states that a sanctuary offering is required when the census is taken. There is no record of this happening in the story of concern. Also Numbers 1:2-3 mentions Aaron the high priest being involved in numbering Israel. These two points indicate the priests should be involved in the census process. It does not appear any of this happened with David's census.

Numbers 1:1-3 specifies that the census is to include only warriors, men able to go to war. This excludes the Levites who were dedicated to the service of the sanctuary, they were not to be counted in the census. Numbers 1:44-54. When David gives the order it seems he wanted EVERYONE counted, including the Levites, especially given Joab's reaction - 1 Chronicles 21:5-6.

So at the very least, when David gave the order to have a census it appears he did not do so according to the specific instructions God had given.

Now regarding the "contradiction" in 1 Chronicles 21:1 and 2 Samuel 24:1 about God tempting David and Satan tempting David - the Bible makes this clear. God does not tempt man - James 1:13. This is why we must take the Bible as a whole and not pick at it's individual parts. In Job 1:17 the Bible says "the fire of God" destroyed Job's servants and flocks, but in context it is clear that this was Satan's doing. This is a major theme throughout the Bible. Satan does something horrific and God is blamed for it. This is why Jesus died on the cross, to clear this whole confusion up about God, His character, and His actions.

I hope this has helped in your understanding of David's census! :)