Really all you have to do is have the 1860 presidential election go much differently, which is ridiculously easy to do, since it was such a strange election.
Remember, Bell came in third in that election, in fact fourth in the national popular vote. He only became President because Stephen Douglas, despite placing a strong second in the national popular vote, finished fourth in the Electoral College, and lobbied strongly for Bell when the House of Representatives chose the next President.
You could of course have Douglas not come out for Bell or sit things out, or just change things with the House of Representatives. Bell was elected President by the delegations of just 13 states, to 12 for Seward and 8 for Breckinridge, and this is with all the horse trading.
Or just change the Electoral College vote so that Bell does not finish in the top three, or Seward wins an electoral college majority outright. Remember, Seward wound up just two electoral votes short, and Indiana and California were really close. And Douglas would have been one of the top three Electoral College finishes instead of Bell if it hadn't been for that faithless elector in Oregon. Bell also carried Virginia over Breckinridge by less than a hundred votes. Of course, with a single Democratic ticket, Douglas by definition can't finish fourth and there is a slim chance he can even pull out an outright win by winning all of his own and Bell's and Breckinridge's states. You can also do all sorts of things with the New Jersey electors.
Aren't there enough timelines on this site about a Seward administration? Though several have the Unionists continue to be a major force, and I think that would have been butterflied away.
Btw, though the party's official name is in fact the "American Constitutional Unionist Party", I am puzzled why the OP uses the archaic terminology instead of just the Unionists as everyone calls them.