That is one of those questionable Lincoln quotations.
If the slave states didn't want high tariffs, they could have stayed in Congress and defeated or moderated them.
Maybe it is all about money as you say, but in those days, slaves and slavery were a pretty big investment in some states -- big enough to secede to protect and big enough to fight for.
“I have no purpose, directly or indirectly, to interfere with the institution of slavery in the States where it exists. I believe I have no lawful right to do so, and I have no inclination to do so.......... In doing this there needs to be no bloodshed or violence, and there shall be none unless it be forced upon the national authority. The power confided to me will be used to hold, occupy, and possess the property and places belonging to the Government and to collect the duties and imposts; but beyond what may be necessary for these objects, there will be no invasion, no using of force against or among the people anywhere.” Slavery where it existed was OK. Not paying duties and imposts not so much.