The worst that can happen here is one of two things, I think. Both have easy solutions.
1) The SIL is really embarrassed, and sort of secret resents Lexi for embarrassing her.
SOLUTION: Lexi continues to be friendly and welcoming, and to treat her new SIL-to-be as "one of her inner circle," without ever referencing the error or the email. Should the topic come up, Lexi just says, "Yeah, it can be confusing to people, they just don't expect that to be my actual, real name."
Time will heal that.
2) The SIL decides Lexi is a completely picky person, a deliberate nitpicker.
SOLUTION: Lexi continue to be friendly and flexible, and to naturally show the ways she is accommodating, etc.
Time will heal that, too.