I want to make a hat cake, but I'm uncertain about the brim.

Do you put a big circle of fondant down on your cake drum and build the cake on top of that or do you try to wrap the brim around the cake later?

If you put it down first, how do you assemble/ice the cake without getting stuff all over the brim?

It depends on what kind of hat you want to make. For the cowboy one, I made the brim well beforehand, so it had time to dry. Then I placed the cake on a slightly smaller cakeboard than the circumference of the hatpart and decorated that seperately. After it was finished I lifted it onto the brim, so that the brim stayed clean. A ribbon/border whatever you might call it would hide the part where the cakepart just sits on the brim
Or if you want it seamless, fill and crumbcoat the hat on the cakeboard, which is larger than the cake, place fondant on it and smooth down as one. Don't cut the fondant right at the boarder of the cake, but add the brimpart, which is just fondant