The north got lower prices for cotton through bartering their goods for cotton. Tariffs were on goods imported from overseas.
In effect every time the south bought goods from overseas or traded cotton for them They paid the tariff that went to the governemnt and when they traded with the North they had to get the goods from the north at a price almost as high as the imported goods with the tariff included.
Either way the south got screwed in the deal and they had no other options.
So how did it work? Did the Northern industrialist say, "I'll swap you two barrels of nails for one bale of cotton?" By the mid 19th century we were not a barter economy. Northern consumers paid cash for cotton, and Southern consumers paid cash for the goods they obtained from the North. Northern consumers paid the market price for cotton, same as European consumers paid. Southern consumers paid an inflated price for any protected goods, same as Northern consumers paid. The concept that the tariff somehow impacted only the South is ridiculous. It hit everyone equally, North and South. The region that imported the most paid the higher percentage of the tariff. And it's clear from government statistics and quotes from the leaders of the time that the states that made up the North during the rebellion paid the largest percentage of the tariff. Which stands to reason because that's where most of the people where.