There can be no coup, because South didn't want to rule US. They wanted to secede. If they did want to rule entire US, they'd negotiate when Corwin amendment was proposed. Corwin amendment would be passed, Lincoln would be hailed as man who saved they Union... and South would know it can threaten to secede to get their way in future North-South disagreements.
Probably a majority of southern political leaders and voters after Lincoln's election were not absolutely committed to secession. But it would certainly take more than the Corwin Amendment to keep them in the Union. Basically, all that amendment did was to prohibit the federal government from passing a law abolishing slavery in the states. Hardly any southerners thought that *that* was how the Republicans would menace slavery. They were afraid of more indirect methods, which the Corwin Amendment did not address. Nothing in the Corwin Amendment would have prevented Congress from outlawing slavery in the territories while admitting one free state after another, thus reducing the South to a hopeless minority. Nothing in the Amendment would prevent Lincoln from using federal patronage to build up an antislavery party in the South. Nothing in the Amendment would prevent the Post Office from allowing the circulation of "incendiary" materials in the South. Nothing in the Amendment would stop northern defiance of the fugitive slave law--even if Lincoln was sincere in his promise to enforce it. Southern fears of all these things may have been overblown, but they were genuinely felt.
The question is what would southerners who were not "secessionists per se" accept as sufficient reassurances to lead them to stay in the Union. At least in the Lower South, the answer was probably that nothing less than the Crittenden proposals could produce an anti-secessionist majority. But the so-called Crittenden Compromise was so one-sidedly pro-southern (especially in guaranteeing slavery in all territories "hereafter acquired" south of the Missouri Compromise line--which could mean anything from the Rio Grande to Tierra del Fuego) that it is very unlikely that enough Republicans would support it to give it any chance of passage. Indeed, many of them argued quite convincingly that it was not a real compromise at all but simply a surrender to the South. (The only "pro-northern" feature of the Compromise was banning slavery in the territories north of the Missouri Compromise line--where nobody expected slavery to take root, anyway. Oh, yes and the Compromise did call for laws to suppress the African slave trade--but so did the Confederate Constitution!)
It is not really true that the South did not want to rule the United States. Or at least it is not true of all Southerners. Even in South Carolina, James Hammond explained why he thought staying in the Union was safer for South Carolina than attempting "go it alone" secession: "the South...can, when united, dictate, as it has always done, the internal and foreign policy of our country." (Note that Hammond is here admitting one of the Republicans' main allegations--that the South, far from groaning under northern oppression, had hitherto dominated the country.) Hammond explained that "at the North, politics is a trade." The spoilsmen "go into it for gain." (This was a typical South Carolina aristocratic view of the "mobocracy" which was seen as prevalent in other states, and especially in the North.) For that reason, no Yankee has "ever been twice elected President." Mr. Lincoln's administration will also break down "before it can accomplish anything detrimental", for its "antislavery agitation" will "not gain them spoils and power." (Quoted in William W. Freehling, *The Road to Disunion, Volume II: Secessionists Triumphant 1854-1861,*, p. 405) No doubt many southerners agreed with Hammond that a southern-dominated Union would be preferable to secession. Where a crucial number disagreed with him was whether it was *possible.* But if it was not possible through political means (including threats of secession) it would certainly not be possible through a coup d'état.