Hi J.,
What about a baby carrier (sling or Baby Bjorn) for your newborn? That way she'll be right up next to you, and you don't have to worry about where to put her on that gross bathroom floor. My daughter loved to be in one, and I carried her until she got heavy enough to start hurting my back.
This seems to be a stage that kids go through. I remember my nieces always wanted to go to the bathroom whenever we went anywhere with them at that age. They're bored, and saying that they have to go to the bathroom gives them some control over the shopping trip, not to mention they get to explore other areas of the store besides the boring clothes that mom is looking at. My oldest niece always used to tell her mom she was thirsty as soon as she got in the car, then drink down the entire contents of whatever her mom gave her to drink, and of course she then had to go to the bathroom as soon as we got wherever we were going. Of course it created some funny memories for us too! I can remember taking her to the bathroom in a restaurant; I went into the stall next to her. In a loud voice, she asked "Are you going poop?" "No." "Why not?" "Because I don't have to go." I got some funny looks and smiles from the people at the table closest to the bathroom door when we exited the restroom - I'm not sure if it's because my niece is so cute or because they heard our exchange.
How about taking your daughter to the bathroom before you get in the car? Tell her that this is her opportunity to go if she has to go, and if she doesn't, she needs to hold it until you get home (or to the next store)? Or at least ask her if you have to go before you head out to the car, and tell her she's going to have to wait and hold it if she doesn't go now.