HI! My DD is going to be 2 in January, as well! I know how you feel about trying to get anything done! Wow!
Anyway, my DD LOVES to go shopping and eating out, but I NEVER let her out of the cart/chair. Once that seal is broken, there is no going back. (Learned that lesson the hard way!!!!) So, timing is probably the most crucial thing. Be sure not to take her to a restaurant with a long wait, or one that takes a long time to get the food. Have plenty of special, restaurant/shopping only toys in your diaper bag.
I have always used a shopping cart cover that is very cushy. So my DD is very comforable. I also use it in the high chair when we eat out.
If you already have one, then maybe you should try to get a really girly cart cover and a soft pillow with feathers on it or something. Make a big deal out of her special shopping/high chair cover and pillow. Tell her only Princesses get to use those, then point out other babies riding nicely in the shopping carts. I don't know if this will help, but they do understand a lot at this age. If she likes to 'do her makeup', maybe you should carry some lip gloss with you and use it (or whatever she likes) to get her excited about shopping. Ask her if is she is ready to go shopping, then tell her she needs to put her lips on, and make a big fuss about what a good girl she is. Be sure to reapply the lips as needed to keep her feeling good about it. :)
I hope that helps. Keep us posted!
(Oh yes, I happen to be one of the Moms who raves about The Happiest Toddler on the Block! I like the DVD, in addition to the book, because it is really hard to use the techniques without actually seeing them in full swing. You feel silly at first, but it really does work.)