What to do with butter that has been mixed with baking soda?

Honestly, if you really don't want to throw it away, I would suggest sticking it in your freezer, and slowly using it up when you need baking soda and don't mind a bit of added butter. Baking soda is water soluble, so the butter shouldn't change it too much or activate before its time, I hope.

Maybe you can mix thoroughly before freezing, and separate into chunks or portions based on about a teaspoon of baking soda - that will make it easier to measure (or guesstimate) the baking soda portion into recipes, and either adjust any fats... or just leave it be, the additional butter may not be enough to upset the recipe, most recipes I can think of that need a teaspoon of baking soda can absorb up to a few tbs of extra butter without much difficulty. It's just much more likely to find baking-soda recipes that don't mind extra butter than finding places you can use butter that don't mind the extra baking soda.

Alternatively, you might take Joshua Engels's suggestion and trying to make ghee out of your butter - it would probably settle out with the milk solids and be left behind, though I couldn't swear there would be no residual taste. Probably best to keep any ghee produced this way for very aromatic or spiced dishes to cover up any minor off flavors. This will likely kill your baking soda, though, between the heat and the water content of the butter (before it is clarified) and the milk solids getting mixed into the residue, so it's still a waste of one ingredient.

As for your cookies, you should probably just go get some more butter. I don't think you can salvage the butter for this batch, or icing, or any other purpose that puts butter first - the amount of baking soda is just too much, the butter will almost certainly taste off and likley ruin any ingredients you're using to try and cover it up.