NuMa maybe - descriptive enough, if you think about it.
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I’d be wary of any mom-related term. My dad and his siblings call their stemom “mom” and “mommy”, but then, their actual biological mom died over 50 years ago. Widowhood, not divorce, is the reason why they have a stepmom. And I guess after so many decades, heck, she’s their mom.
From my experience, I’m partial to auntie (tití). It establishes a similar role in raising the kid, gives her a family title, but is not related to the word mom in any way (and the mother is still very much a part of this kid’s life).
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Not unless she likes …buttons…
Milady?
I too like Mama Ray, but if she doesn’t want to add a Xtian name, what about Mim? Close enough to Mom, but differentiated enough so that TLB’s actual mum shouldn’t get her nose out of joint too much.
How about just M? (edited to add) M, Em, Emmy, Emmy Ray. It lends itself to creativity.
My friend’s stepson just treated her name like a title. “This is my Mummy, this is my Daddy and this is my Wendy”.
Maybe TLB and Ray could come up with some completely non-mom, not plain old first name name to use that’s special between them.
When my son was little he called his father Dad or Daddy, and his stepfather, whose name is Robert, Harry. No one else called him Harry, it was just between them.
Mommy Ray, Mama Ray, preferably choosing a title which is not one used with your first wife.
I’m from a culture where “stepmom” has very negative connotations and calling one “Mamá [Name]” has become pretty much the norm. The original Mamá doesn’t get a name, it’s not needed.
In the UK (of my imagination) kids call their mothers “Mumsy”.
My lesbian sister’s daughters call their parents “Mom Fran” and “Ma Mary.”
I hope this qualifies as IMHO, but ‘bitch’ was the clever word I had for it.
In your situation, either “mom” or the woman’s first name is appropriate.
How you tell TLB that it’s polite to call the person what they want to be called…
What does Rachel want to be called? I think she and TLB should work something out together. FWIW, if they both like “Mom” I think they should go with that. Definitely not mother because like you said, she has a mother. If she’s living with you and Rachel full time, and Rachel will be doing a lot of “mom things” with her, it sounds like Rachel has earned the “Mom” title.
Demi-Mutter
or
Demi-Vormund which is German for “guardian,” plus “demi” because it sounds cool and cuts it in half.
A friend called her step-mom Esme, which stood for Evil Step MothEr, but in an affectionate way. It wasn’t her real name or even close to it. But she was older (late teens, early 20’s) when she first met her stepmom.
Aaaak. No, they don’t!
Mumsy means ‘frumpy’. Not a name any woman would want.
IF Mama Ray is too Southern, how about Ray-Mom?
Maybe she can call her “Love” 
It seems like you have a lot of that in your home.
How about Miz Rachel?
My kids have a stepmother and I was thrilled when they were close enough to her to give her a momlike name. Talk to your ex and explain that TLB is looking for a name to call Rachel and that since she is Mommy TLB would like Rachel to be Mom. Don’t blindside her with it but you can’t stifle your daughters relationship with Rachel to make her happy. If she’s unhappy with Mom let her make suggestions but it’s got to be clear that TLB is looking for a mom like title.
Mom 2 Electric Boogaloo.