I think the idea is letting the pets sniff the blanket, not giving it to them as a chew toy. I'm strangely puzzled that you think your pets so unsanitary that even after washing a blanket they had come in contact with that it would have be thrown away.
It's just the thought of having their noses on it and if they lick it or anything like that just gross and it bothers me and I wouldn't be able to get that out of my mind and wouldn't ever be able to put back on my baby. I'm just that type of person, many things don't bother me but some really do and this is one of them. I am constantly going around house cleaning and sterilizing things. I'm really OCD about things like this.
You can use whatever you want to that has the baby's smell on it... the hat, like mentioned, a bib, some socks... snitch a receiving blanket from the hospital and have the dogs smell that?
Honestly though... we didn't do anything like that at all when my son was born. We brought him into the house and they were fine. They know that he's a "human puppy" and were very wary around him. They also could see that we were very protective of him. We let them have a good smell of his feet and blankets and stuff when we got home, and then they spent the next few weeks adjusting to him, just like they would have to even if they smelt him before he came home. We'll be doing it the same way this time!
The way I figure it... its not safe to leave your baby alone with your pets anyway, so I fail to see how "introducing" them to the smell beforehand is going to make a difference. So long as you never set baby in the bouncer or on the playmat on the floor or whatever and then walk away while your dogs are capable of getting to the baby, then it shouldn't matter. Our dogs were only allowed around the baby when we were holding him, and they never tried to lick him or anything (and we have VERY lick-y dogs).
I would carry him with me, set him in the bassinet or in his bouncy chair, close the bathroom door behind me, pee, and go back out.
If, for whatever reason, (say, he was sleeping in his swing and I needed a bathroom break), I'd take the dogs with me and make them sit in the bathroom with me. I'd lock them in the room with me while I was on the elliptical so that he could nap in his swing. They're never going to attack the baby while you're holding it, so its perfectly safe to introduce them while you're holding the baby. Everything else seems like an unneeded extra step to me.