I feel like the traditional way of tying off on a cleat does not work on these larger split cleats,
Yes, it works. Our standard cleat hitch -- on any horn cleat -- starts with a full 360° round turn, though... then figure 8 to lock. Don't know about starting with a 180° round turn, if you're doing it that way; we just don't do that.
You probably know, but differences are illustrated here:
How to Cleat a Line: Three Ways | Expert Advice - YouTube
You can also lay a pre-spliced cleat through the split, and then return over the ears, ditto like working with a standard horn cleat that has a "hole " under the middle.
-Chris