Hi there!
Sad news, there is no cheap way or loophole to buy such an expensive bracelet.
The only way to avoid sales tax is to not live in a state that has a Cartier boutique and purchase online. And technically even then you're supposed to declare it and pay the sales tax you owe your state
I was able to purchase online and not pay the sales tax but paying the tax would not have deterred me. Yes it's another couple hundred dollars but you've already justified the several thousand dollar bracelet, so...haha!
You could always shop around for a "pre-loved" Love bracelet but with that always comes the thought in the back of your mind that it COULD be a fake....just look at all the posts and threads on this forum of people worrying themselves sick over paying thousands of dollars for something potentially fake. Used they are often basically just as expensive as new because they hold their value well. Honestly it's best to just bite the bullet and buy it directly from Cartier if you've come this far.
I can't answer the Heathrow question because I've never been and I don't know how that works based on what terminal you're on/the timing! I've always been curious actually.
EDIT: Maybe see if she'd consider the newer, "small" size Cartier love bracelet. It's a little more than half as wide as the original, and for yellow or rose gold comes in at 4,050USD rather than 6,300USD. That's what I did because I liked the daintier style anyway, and it looks just about the same as the original, especially if you have very slim wrists like I do. Plus, I'm not made of money and yes please to a 2 thousand dollar savings haha!