You figure the best defense is a good offense. If you can make it look like you're just wearing a costume, people might give you strange looks, but they won't dig deeper. They might just think you're nuts, but it's better than them realizing you are now part dog.
You consider carefully where to actually spray yourself with BOD. Fur, reiterating your prior thought, could be shaved, but bodily changes had no guarantee of reversal. You need to protect the areas of your body which you definitely do not want changed. Setting down the spray, you quickly put on some long gloves and socks, realizing that having only paws and no opposeable thumb could severely limit your ability to function. In order to make it look like a costume, you want to perform the absolute minimum of changes possible, since you don't really know how the spray will affect you over time. In the end, you have decided on two changes which should provide enough of an effect to back up your costume story.
First, you find an old ski mask. You put it on after cutting two small holes in the appropriate locations. You then don a pair of sunglasses and a face mask. You take a deep breath, hold it, and then resolutely spray the only part of your body now visible outside your clothes: your ears.
At first it feels almost like you have water in your ears, and you think it's just that you sprayed too much BOD into them. However, the sensation changes, almost like ants are crawling in and out of your ear canal. You resist the urge to scratch, and soon you are aware of a new problem. The pressure from the ski mask increases around your ears, suggesting the holes are too small, and they are holding your ears in the wrong position. Immediately you drop the container of BOD and rip off the sunglasses, face mask, and ski mask. The feeling of relief washes over you, but the strange twitching around your ears continues.
You raise your hand to finally scratch, but find your ears are no longer on the sides of your head where they have been all your life. Instead, they are set higher and stick up out of your hair to perky, furry points. The urge to satisfy your itch is more insistent, however, than analyzing your new anatomy changes, so you reach up and scratch behind your new protruberances. As you scratch, you feel every muscle in your body fill with a sense of bliss, and you find you are panting open-mouthed in tongue-lolling pleasure. A little shocked by this, you walk over to the mirror to inspect the alterations. Your new dog ears are quite furry (which explains the itching), and are also easily moveable, as you find when you hear a strange noise outside. Your ears turn slightly almost involuntarily, and stand up straight and stiff. You find they are also more sensitive when you peek out of the window to see that the sounds are the footfalls of your neighbor across the street walking to his mailbox.
Dragging yourself back to reality, you recall why you changed your ears in the first place, and decide to move on to the second stage of your plan. The other change you will make in order to give the impression of wearing a costume is...