if Andrew Jackson was such a champion of states' rights, why did he act so harshly during the Nullification Crisis?

For starters let's consider if Jackson acted harshly during the Nullification crisis. While we commonly focus on the Tariff of 1828 as the instigator of the crisis, it should be remembered that a compromise tariff was passed in 1832. Drafted by John Quincy Adams, the tariff of 1832 was a compromise tariff that enjoyed wide support both in the North and the South and included a reduction on many rates from the almost absurdly high rates of the 1828 tariff. This tariff came with the blessing of Jackson but was also nullified by the State of South Carolina. It is only after the initial attempts for compromise had failed that efforts to enact the "force bill" were enacted.

As to why he treated South Carolina more harshly as opposed to other state's rights efforts, there a number of factors to consider. For starters none of the other state's rights efforts so blatantly challenged Federal supremacy as Nullification. Many of the other state's rights initiatives were also spearheaded by Jackson, Maysville road veto, recharting Botus II, and Indian removal although often with the support of state righters. The other factor to consider is that Jackson hated Calhoun by 1832/1833, by the time Calhoun resigned his position as VP Jackson began to look on the Nullification crisis as a direct challenge to his authority and a contest between Jackson and Calhoun. Challenges to Jackson's authority had played out before, namely in the Eaton affair which resulted in the mass resignation of Jackson's cabinet.

I could understand trying to stop South Carolina, but threatening to invade them

Very few political figures believed it would actually come to war, or that South Carolina would secede. Jackson played his hand brilliantly. For his faults it is important to understand that Jackson was a very intelligent politician. He passed a number of state's rights proposals that caused South Carolina to be isolated in their challenge to federal supremacy. He reinforced the forts in South Carolina with federal soldiers, and moved the collection of revenue to armed revenue cutters and the forts themselves. He ordered General Scott to make plans to invade South Carolina, and publicly threatened to hang any nullifer who laid a hand on a federal official. When McDuffie (South Carolinian who lead Nullification in the house) quested if Jackson would really hang anyone, Thomas Hart Benton (Jacksonian spoke person) pointed towards his execution of British agents. Jackson's reputation for violence and military action here played perfectly into the Federal government's hands. While Lincoln's actions nearly three decades later were unknown or murky at best, Jackson's actions were blatantly clear. However let me point out that Jackson ordered federal forces not to fire the first shots, if a war was to come South Carolina would be the aggressor.

Additionally, what other achievements of his were such profound victories for states' rights that he can still be considered a champion of states' rights even after his action during the crisis? Thanks for your help.

Censure of the mail, Indian removal, blocking rechartering of BOTUS II, vetoing Maysville Rd. Jackson can still be considered a State's Right supporter because being a Unionist and supporter of state's rights are not mutually exclusive. The Democratic party was built on this basic ideal and would play out again under the presidency of James Buchanan. Moreover there are varying degrees of support for State's rights, Jackson for instance denied the right of a state to secession ( a contentious issue in the day).