You can find the Porter Outfit in the Porter Closet. Put it on and go to Tesla's cabin. He will let you in.
When you're in, pick up the jug of prune juice on the table, then click on Tesla. He'll ask you three times to fill his cup, then he will need to go to the bathroom. While he's gone, the item you click directly before he enters will be where the key is. Then Tesla will come back, and he will tell you to leave.
If you directly customize this outfit from The Porter, you will not be able to get past Tesla.