Can I convert this cocoa frosting to a coffee frosting instead?
The first thing I notice is that you are missing starch. Cocoa powder is 60% carbohydrates by weight. It will change the texture a lot, giving body and smoothness. The rest is fat and protein. 

I would suggest that you start with 60% as much starch as your usual cocoa, and 5 tablespoons brewed coffee instead of the water. Cook your pudding - you may have to cook it through instead of using the double boiler, I don't know if cocoa starch is behaving differently, but when adding cornstarch to custard, it is normally cooked through because 1) it thickens better and 2) the yolks can "eat" your starch within a day, liquefying it, unless deactivated by heat. You can try small batches with both versions and see which one turns out better. 

The fat in the cocoa is not so much, so maybe it's not worth replacing it. But if you want a fat which gives the whole thing a bit more solidity, add cocoa oil, about 15% of the original cocoa powder weight. Or even add more of it and reduce the butter for a stiffer effect. 

Using your original amount of sugar and Arabica coffee should result in a recipe which isn't too sweet. But if you still have problems with sweetness, try adding small amounts of acid, maybe cream of tartar (citric acid has a noticeable flavor of its own).