Hi Paul, I can understand why you would feel disrespected by your kid. You asked her not to do something, and she did it anyway. And in front of someone else. Not cool.
Yes, pre-teens and teenagers will be just that

it's about testing boundaries and questioning EVERY little thing (for the most part), so it's wise to pick your battles. I think when it comes to respecting you, your decisions and decisions about money, this is worth working out. How else is your daughter going to learn?
It's my belief, parents should stand strong and firm together in their decisions; otherwise, the kids aren't going to take you - or your marriage seriously (it's all about teamwork and setting a great example for the kids, so they grow up to be lucky enough to have the same type of relationship).
It's certainly NOT to be portrayed as you two against the kids type thing at all, though, either because that will really affect the kids self esteem and feelings of self worth.
You and your wife should definitely discuss the matter in private (when your daughter isn't around), so you can come to a mutual understanding that benefits the welfare and well-being of your kid. When you are both calm, speak with your daughter.
You are the parent. She is the child. It's important to set boundaries and not allow children to manipulate one parent, so it ends up being good cop / bad cop scenario every time the child doesn't get what s/he wants. Like this situation....I think it's setting everyone up for disaster.
Best of luck! Please let us know how it goes!