You've got it backward: The Constitution does not authorize tarrifs that are 'protectionist in nature' unless they fall into one of the mentioned categories:
1. Pay debts.
2. Provide for the common defense and general welfare.
The Whigs (and the Republicans that followed) had no pretentions that the tarriffs they would impose had to do with either of these things. The Constitution musn't just not prohibit something, it must specifically authorize it.
Read the Morill tariff . The legislation provided for the payment of outstanding treasury notes and authorized a loan in addition to fixing tariffs. That seems to meet your definition of the Constitutional purposes of a tariff and disputes your claim that the Republicans made no pretense of it being for any other purpose. Tariffs were implemented first and foremost to fund the government. They had, as an additional purpose, the protection of domestic industries, including the ones in the south, but did not in any way violate the Constitution.