Bitcoin API with mixing service
Check out Bitmixer.io, it even has an API.

You could do that, but personally, I think it's a much better idea to generate a new wallet per customer transaction. That way, all your coins aren't grouped together, so the customers won't know how much you make. Just how much they paid. Personally, my app creates addresses through the Coinbase API, all of which are held in a specific account named after my app.