Pretty much any solid fat will work here. Since you're replacing something that's not just fat, you might prefer to whip the fat to lighten it up a bit, no matter which you choose.
Since your recipe already has coconut milk, coconut oil seems like a really obvious choice - it's common in recipes like this, especially vegan/paleo ones that want to avoid butter. Beyond that, you've excluded butter/margarine, but that still leaves shortening and palm oil. Shortening is one of the standard ways to make frosting (so maybe that means you're not interested?), but palm oil is pretty common too. (Some people may not like palm oil flavor-wise, though.)
There's also cream cheese, which again adds some flavor, but people like it in frostings, and you can find plenty of actual recipes using it, rather than trying to substitute in this one.
Cocoa butter, as Fabby suggested, is also an option, though along with having to add oil to get it soft enough, it's also pretty expensive. It's also maybe not worth it: you'll be adding cocoa powder and sugar to it, so why not just use actual chocolate? You can always use a combination of regular and white chocolate, if you don't want too much chocolate, and again you can find plenty of recipes for this kind of thing. But if you do want to try cocoa butter despite that, note that you'll want to try to test it at a reasonable temperature: just barely melt it so that you can add oil and mix well, then take a bit and cool it to see if it's soft but solid at room temperature. Borrowing a trick from jam-making, putting a small amount on a cold plate is an easy way to test.
The other main common solid fats are animal fats - lard, (beef) tallow, and so on. They'd work too, if you liked the flavor.
I could see peanut butter (or other nut butter) also working, if you like the flavor. For the firm, processed kind, you could probably use it directly, but for the softer kinds, you might have to combine with a more solid fat to keep your frosting from getting too runny.
Also, since your recipe already has coconut milk, increasing or decreasing the amount is another good way to adjust the texture, just as with basic frosting recipes you often add milk until it's right.
In that same vein, you could probably use pureed/mashed fruit to adjust, and even provide a little bulk if your fat is firm enough. Banana might be an obvious choice, since it mashes smoothly and easily.
Finally, if you really want to avoid "normal" solid fats, you might want to look away from normal fat-based frosting and toward things like bavarian cream, thickened with gelatin instead of just fat. That's more of a different recipe than a substitution, though.