Starting from the top: If May is anywhere near as clever and convincing as she is presented as being, she could easily have had her man and kept her position in society. If she decided she had such contempt for society that she wanted out of it, then I have to ask what her rationale was for not marrying the man she picked. I'm by no means suggesting that women didn't do this -- but having recently read _Living in Sin_, I'd argue that women who chose NOT to get married in this time frame usually did so strategically. And there was no obvious strategy here. If you're thinking of arguing something about independence, well, plenty of women wanted independence, but usually they figured out that the problem wasn't really the _man_ per se, it was the kids, and knew not to have them (or to run away and leave them behind, but that's unsympathetic so let's just not go there).
If May and her chosen man were so poor that she had to work, what did they do once she had the baby/babies? Who was taking care of them while she was working/how did May eat and feed the kids, if the man was off drinking his wages, such as they were? Having read _A Social History of Wet Nursing_, I like to think I have some sense of what's involved in keeping the kiddies alive while poor and needing to Not Be There, and this was sort of blown right over with no explanation.
Finally, while there really were genuine problems at Dunning [ETA: I know he calls it Lake Forest Lunatic Asylum, but that is clearly an import from a much later era. The only possible facility I can find for the time is Dunning; Kankakee wasn't open until a little later], it's actually _not_ plausible to believe that May was sent there on the pretext presented in the book. We were not doing shit like that in that time and that place. While you could imagine that you could pay enough people enough money to do what May's mother/parents did to her, your imagination is better than mine. This is a time period in which women who _wanted_ to be incarcerated for mental illness so they could write exposes (and put up a very convincing show) didn't actually have that much trouble getting back out and publishing their exposes.
I'm treating the basic idea of sending 1000 White Women off to make babies as a McGuffin and not getting into the motives for actually doing that. That said, "contractually" only obligated to have one baby? If May or Martha or whoever had walked out and said no I'm not going to, was someone going to _sue_ them? And what would be the recourse?
Yeah, I don't think so.
I have a lot of problems with alternative histories in general (and as a long-time reader of speculative fiction, I've slogged through more than my fair share), most of which can be summarized simply: no one alive at that time in that place would ever have made as stupid a decision as the author has them make for the purposes of the story the author has decided to tell. Whether it's trying to publish a newspaper in the 400s on vellum (actual skin! newspaper! on _skin_! The mind boggles) or the wild series of decisions that goes into the setup of getting May Dodd out on the prairie, I'm far too literal minded to go along for the ride. And I say this as a person completely prepared to accept FTL travel for the purposes of a good space opera. Feel free to break the rules of physics, but try not to frighten me with your inability to make decisions that are psychologically and economically appropriate in the environment around your characters. They don't have to be _good_ decisions, but you don't get to parachute stuff in from another century (or millenium).
ETA: Also, white man writing a book from a woman's perspective about Native Americans in a way that sort of makes the whole story a "look at the cool secret history in my ancestry" feels a little colonialist. But that might just be me.