Lincoln actually came out in support the original proposal of a 13th Amendment. The South rejected it, because to them, because it didn’t address the issue of taxation, tariffs on imports and exports.
Here is the text of the proposed amendment that Lincoln supported:
No amendment shall be made to the Constitution which will authorize or give to Congress the power to abolish or interfere, within any State, with the domestic institutions thereof, including that of persons held to labor or service by the laws of said State.
https://www.gilderlehrman.org/content/proposed-thirteenth-amendment-prevent-secession-1861
Lincoln didn’t support the Corwin Amendment - he only said that he had “no objection”.
The 13th Amendment was passed out of Congress and sent to the states long after the Southern states announced their secession and adopted a constitution that protected slavery to a far greater extent than the 13th Amendment would have. When you say the South rejected it do you honestly think they would have come crawling back for any reason, much less tariffs?