I'll number my comments because this is a complex, multi-part question.
1. Pretty much anything and everything can butterfly away the American Civil War. Just about everywhere else, the only exception I can think of being Haiti, slavery was abolished peacefully. And the nineteenth century for some reason was a particularly bad time for secession or regional independence movements. They were tried elsewhere, but you don't here much about them, since the Confederates actually got much further than the norm.
2. The closest analogous example is Brazil, which was a monarchy that had both slavery and secession movements, and an active abolitionist movement.
3. The personality of the monarch or monarchs makes a difference.
4. One reason why people were so happy to craft a fairly powerful presidency and to make Washington the first President was precisely because he had no natural children and couldn't found a dynasty.
5. How the monarchy happens probably matters. Getting a monarchy in the first place is difficult.
a) If the idea is to make Washington the first King, the framers probably adopt an elective monarchy as in Poland. This looks alot like the IOTL presidency except the term in office is for life.
b) They could try importing a dynasty from Europe, which was discussed, but the problem is finding an under-employed Protestant (and he would have to be Protestant) Prince who is culturally compatible and can produce heirs. I suspect this would have gone over as well as the early attempts to impose monarchs on Mexico and Greece.
c) A third option is a Hanoverian collateral dynasty, either as part of a negotiated peace deal with the UK after a better military performance for the UK in the ACW, or the British set this up themselves (actually an independent friendly American state with a relative of the British king ad the king would have made sense for them, but this was never considered). The inability of the sons of George III to produce legitimate heirs becomes a problem here.
d) A fourth option is what wound up being done with Canada, but this is more than somewhat anachronistic, and now who is the Governor-General.
e) The fifth option is the Young Pretender, Charles Stuart. He isn't demoralized by "the 45" or it doesn't happen, kicks his alcohol habit, offers his services as a general to Congress, is accepted, and winds up having a good war record. The record is good enough to overlook his Catholicism and he lives longer than IOTL. However, his heir is still a Cardinal.
f) Napoleon escapes and flees to the US in 1814 instead of going to Elba. He somehow gets command of some American forces and a good interpreter and does well with him. He settles down, lives longer than IOTL, is naturalized, and gets elected President in 1824 with the "natural born citizen" thing being ignored. He then gets installed as monarch just before finally dying. His heirs are whichever descendants of one of his brothers who have also established themselves in the US. Also, the Bonapartes are Catholic.
As you can see re comment # 5, all of these options are really fanciful and lead to lots of butterflies. I suspect that an elective monarchy is most likely, in which case not much changes, just presidential, I mean royal, elections become really really important. The Adamses might try to establish themselves as a sort of royal family. With all the others, except for maybe the really fanciful d) and f), I think the dynasty winds up collapsing by mid nineteenth century. The monarchy thing really should be its own thread.