I'm a branding professional and I have done this for the past +10 years. For a project like this you should definitely charge a fixed price and bill according to the client's perceived value of the income, that I imagine will be pretty high since we are talking about the branding of his business.
Usually these types of projects take longer than anticipated because with branding and naming there is a lot of back-and-forth, since this is something that clients like to be very involved with. Therefore, I suggest to calculate what you expect will take you to complete the project and add a contingency that could be a minimum of 20% more. I don't want to give you an exact number in US$ because I understand costs in other countries are very different so in % you should visualize it better