How do I add a split button without using the "a href" markup?
You can't. The second href is what the listview widget uses to determine that it will create a split button.
Well, OK, you can, but it is a hassle. You could duplicate the HTML that the listview would normally create for the split-button. That might work.
If the way the listview works isn't what you want, I suggest you create your own list, and leave the listview behind You will get much better performance, to boot.
I don't want to use href because I don't want to open a new page or to
show a popup. Just a simple select menu like this would do:
The select menu has to get there somehow.
How were you planning on making it appear?
If you just wanted the menu to always appear (in place of the split-button icon) then that's not a split-button. You could just include that as part of the listview item styling.
But good luck fighting with the listview CSS! Make your own list.