Judging from the original intention of this matter, I think there is nothing wrong with you, your daughter-in-law and both parents' starting points, but everyone has different views and has not received enough respect in the final result. The reason is very simple, that is, everyone thinks that their ideas and practices are for everyone's good, but they can't get approval and behavioral support all the time, and contradictions naturally arise.
You should think about such a problem. Marriage is a very serious matter, unlike children playing house. Since you can give your daughter-in-law considerable affirmation in the first six months, don't waver in this matter. Everyone with a discerning eye knows that what your daughter-in-law said to you was angry words. Whether she is right or not, she did express some feelings about it. You should consider some problems from her position. The so-called "saving face, her family's conditions are good and she spends a lot of money" and other factors should not be the reason for your current vacillation, otherwise what would you have done? As a man, you should have absolute control over the relationship between mother-in-law and daughter-in-law. Don't back down just because you can't handle it, or even if you escape this time, can you promise that it won't happen again?
Finally, tell me about your wife. According to your description, this girl is really spoiled. You can't be in a hurry to deal with a person of her character. It is impossible to reverse her long-formed nature by one means. It is still necessary to deal with some things patiently and use some methods and means wisely. Being too hasty will only be counterproductive.