The South didn't fund the government. Tariffs did, and those tariffs were mostly collected by goods coming to the northern states.
I don't see how you think the North paid lower prices for cotton than the rest of the world, except for the lower cost of transport. Cotton was untaxed, like all exports.
Imports were taxed via the tariff.
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.