Doesn't really matter whether you chill in the kettle or in the fermenter, and as some point out, if you chill at all. There's always waiting til the next day and let it cool naturally. There is some infection risk in open-air fermenters doing this, but if you can blanket it with CO2, you're golden. Just gotta keep the baddies out.
I'm gonna second what the others are saying about hauling hot wort around. You're flirting with disaster. I'd find a way to cool in the kitchen, somehow. Whether in the fermenter or in the kettle, doesn't matter. Carrying hot wort downstairs is a horror waiting to happen.
Similar issues made me move my operation out to my tractor shed. I have 14 steps to my basement, either inside, or outside, which is a LONG flight of stairs when you're carrying something as awkward as a FastFerment 7.9G fermenter. Now all I have to do is carry the bottling bucket inside after I drain the fermenter. I'm not yet set up for kegging or washing bottles out here, but the bottle washing will come out here soon. Once that's done, I don't handle anything hot more than a few feet, and even that will likely be via pump.
It's a fun hobby/craft, but face it, a nasty burn from it would get un-fun very quickly.