I have a similarly stubborn, power-struggle inducing DD1, who's 3.
I can totally understand what you're saying about her slowing down even MORE when you try in the slightest way to speed her up, and I totally get how infuriating and draining that can be.
I think you know her best. I think it's a good idea to take her to the ped, even if it's just for her to feel like you're listening. Have you thought about taking her to a chiropractor?
I'm wondering if this might be a self-esteem issue, if she is indeed faking it. It seems attention seeking. It seems designed to get the pity/attention of others, as well as to push your buttons. I might work on my own language toward her, building her up about things completely unrelated to the limp issue (ie, not talking about how great she did when she went somewhere without slowing you down, but instead talking about how lovely her smile is and how it really makes you happy to smile with her, and how you love to hear her sing, or what a beautiful drawing she did and it'll go on the fridge for Daddy to see, etc.). I know with my DD1, me trying to hurry things along (especially if I forget and imply that we're late or need to hurry) just makes everything take a gazillion times longer with her. If I know we're late, but don't mention it or push her along more quickly, we can make up time.
I really like the idea that when you're out, and she says she needs to pee, you say something like "OK, and when we et to the bathroom, we're not leaving until you have gone pee."
I also agree that a movie in a theater was likely too much for her. I know there's no way my 3 yo is ready for a movie in a theater.
I wonder how things would go for you if you just slowed down to her limping pace. It would drive me batty because I have no patience for DD1's dawdling, but I wonder, if you spent a day or two intentionally going at her pace, and talking about all you see as you go along about your day, and just generally allowing for a slow 3-year-old pace, if she'd be OK with moving more quickly in a few days.
I'm glad you're taking her to a ped and a neurologist based on the ped's rec., I think it shows her that you think this is a big deal, and if there's a physical basis, you'll all be happy to get it sorted out. Try to ignore Dr. Google (love that!) and don't waste any more time on that, it'll likely just make you even more anxious.