I will first look at your three scenarios:
I could gift her $37,000 and ask her to pay it back later. But that
exceeds gift limits and also seems “wrong” as it isn’t a gift.
The lender will notice the $37,000 deposit into her bank account, and want an explanation. If the story is that it is a gift, they will require you to sign a form stating that it is a gift and it doesn't have to be repaid. Swearing it is a gift when it isn't is loan fraud.
Yes it exceeds the annual gift limit. But that means the excess amount comes off your lifetime limit, and that limit is in the millions. There will be paperwork for the IRS.
I could loan her $37,000 with some kind of formal agreement. I’m not
sure of the tax implications of this (and I’m not sure if her lender
would have a problem with this but I can ask.)
If you collect interest, and you should, it will be taxable income for you. The lender will want to know what the monthly payment will be. The obligated monthly payment will be factored into her ability to afford all her required payments. If they accept the plan that the CD when it matures will retire the loan, the harm may be minor. But the question is will they believe it. Or will they wonder if the loan will continue for years because the CD will be rolled over.
I assume I could “buy” her CD from her for its current value. Not sure
if this is possible.
I do not know if this is possible. I know in theory it is, but I have no idea if in this specific case this can be done.
There is a 4th possibility.
Cash the CD is in early. There is a cost to this. The paperwork connected to the CD will define the early withdraw penalty. Generally it is stated as days or months of interest. Depending on when the CD was purchased, and the size of the penalty it can mean that the amount returned will be less then the purchase price. For example if there is a 90 day penalty, but you cash in after 30 days.
The fine print needs to be studied. The calculation needs to be made.
The CD doesn't need to be cashed in until just a few days before settlement. That means it might collect a month or two more interest before it must be cashed in.