I think even with IVF, a court is not going to rule in favor of Kathryn. Contracting for a child (barring gestational surrogacy) is not allowed here. South Carolina does not look favorably on anything that could look like "buying a baby," which a skilled attorney could probably successfully argue that Thomas was essentially paying Kathryn for the baby. Moreover, there's something called the statute of frauds, which would almost certainly apply if you couldn't get it kicked on public policy--their agreement would have to be in writing. As for promissory estoppel, Kathryn getting pregnant would likely never be considered "detrimental reliance" because it would set a bad public policy example.
She's too immature and selfish to see that the best way to get what she wants is not to act like a crazy person. I live here and have heard he offered her much more than $2,500 for an amicable agreement about child support and she went batshit crazy on him so he withdrew the offer. Unfortunately for her, he has the money and thus, the power to control whether he wants to give more than the maximum. The fact that these two don't have a settled child support order is crazy to me and suggests that Kathryn does not have an attorney. Any attorney worth their salt would get in front of judge ASAP.