The new CSA Constitution itself did not give the CSA states the right to secede but did the CSA Constitution preamble?
Don't know why that would matter; it has nothing to do with Lincoln's deliberately starting the war.
We do how ever know that Madison rejected giving the federal government the power to use military force to suppress a state that seceded, he scotched a proposal to do just that during the Constitutional Convention, and with Jefferson co-authored the Virginia Resolution of 1798 while Jefferson authored the Kentucky Resolution alone. If the attendees of the Constitutional Convention thought it should be illegal to secede and wanted the federal government to have the power to force states to comply by using military force they would have voted for that when it was proposed then. They didn't.
We have the Federalists in New England threatening to secede merely because Jefferson was elected President, and again in 1807 when Jefferson began enforcing the embargo with federal troops, and again in 1814 or so. Obviously they didn't think it was illegal, and since none were arrested for treason for attempting it, even during a war, it's clear Jefferson and his faction didn't think it illegal and treason, nor did Madison when they tried again during the War of 1812. Daniel Webster didn't consider it illegal, since he sided with the New England faction at the time.
As for slavery, slave states were brought into the new government, and remained there. In the 70 years prior no amendment to outlaw slavery was ever seriously proposed, nor any Amendment declaring secession illegal ever proposed. Lincoln didn't think secession was wrong or illegal, he supported states who wanted to secede, as in his speech citing the Texas secession in 1848. Since Lincoln was himself a white supremacist, and help write and pass greatly strengthened Black Codes in Illinois in the mid-1850's right before the war, along with several other northern states, and the Republicans running on white supremacist planks in election right up and during the Civil War itself, it's clear they weren't running on the evils of slavery and the plight of the black people in America, and Lincoln himself saying numerous times he wasn't going to do a thing about slavery, the pickings for those claiming the war was about slavery have pretty much nothing to stand on re the historical records. The only issue Lincoln wouldn't compromise on was his tariffs; they were the key to the rest of the old Whig 'American system'.
We also know how Lincoln's military governors handled 'free' blacks and how Lincoln planned to handle that problem; Butler and Banks in Louisiana ordered 'freed' blacks to remain on their plantations and work at what they did before, and not to leave without written permissions from the plantation owners. But hey, the were 'free' and got paid wages! $3 a month, a wage also set by Banks.
The 'anti-slavery' fans in the North were about keeping blacks in the South and the new territories 'free soil for free white men'; very few were opposing slavery for moral principles outside of Quakers and Garrisonites, and a sliver of Radicals in the Republican Party.