Thank you for writing to Just Answer.
As a stay at home mom with 3 kids, your cousin's wife may feel like the adult "outside" world is leaving her behind. She may be somewhat jealous of your success in management consulting--and when people feel jealous, they often go on the offensive.
I doubt this really has anything to do with how one speaks to customer service people--I think this has more to do with her feeling somewhat insecure around people who are out in the business world while she's home with children.
Rather than take offense or try to get her to stop her behavior, try putting yourself in her shoes and see that this behavior is stemming from her own insecurity. Saying "You may be right, maybe I'll try that next time" will cost you nothing, but validating her on these issues (which are, as you pointed out, mundane and unimportant ) may help her feel less insecure around you, and may eventually extinguish her competitiveness.
You have many opportunities to be validated in the business world..she has to be "sweet" to get things done. I think by taking the high road you will be doing her a kindness, which should help make her competitiveness less irritating to you. Some things are just not worth arguing about, and saying "you may be right" is a great way to avoid conflict with a family member.
Be very clear that I do understand that you know perfectly well how to interact with customer service people, etc. This is just a way to defuse her competitiveness.