No way would the non-western countries put up with it. They would probably refuse to play and if enough of them did, they could gut the whole thing.

https://www.koreaboo.com/news/chinese-female-athletes-go-viral-accused-looking-like-men/

https://filipinotimes.net/lifestyle/sports/2019/09/01/winners-chinese-womens-track-field-look-like-men/

Winning that medal is not just for the individual, it is for the pride of the country. Entering a female athlete into a competition with a man would be accepting the premise as well as condemning their female athlete to lose to an already more powerful and affluent west (that's probably screwed them over at some point, maybe recently.)

But many of the countries you say would balk at "entering a female athlete into a competition with a man" have had no problems entering male athletes into competitions with women. After all, since mandatory sex chromosome testing of female international athletes was abandoned in the mid-1990s, the countries that have made a point of finding & training XY DSD athletes to enter into women's international competitions have all been poorer countries of "the Global South" such as Kenya, South African and Burundi. A major part of their motivation has been to get medals "for the pride of the country," as you say. But that's not the half of it. The other aims involved have been the desire to stick it to Whitey from "the Global North," to challenge the hegemony of the European-run sports governing bodies like the IAAF/WA, and to upset the apple cart, as it were, of binary sex supposedly invented by white Europeans and imposed on the rest of the world through colonialism. All as payback for all the ways the more powerful & affluent west screwed them over in the past.

The central arguments that Caster Semenya, Athletics South Africa and their attorneys have advanced to justify why XY DSD athletes like Semenya should be allowed to compete in women's sports are the same ones that TRAs use to explain why they think TIMs belong in women's sports: an athlete's claimed "gender identity" & what his or her passport says about his or her sex/gender always trump biology; "sport is a human right" - and being able to choose which sex category you compete in should also be a human right, at least for males; regulations that require XY DSD or trans athletes to take drugs to lower their testosterone to compete in women's sports are a human rights violation & the "legal equivalent of pharmaceutical genital mutilation;" no one has ever done scientific studies proving that males have a physical advantages over females in sports, maybe female athletes just don't try & train hard enough; but even if males do have physical advantages in sports, so what? Inclusion, "respect" for claimed "identities" and providing individuals with trans identities and DSDs with "gender affirmation" are far more important than fair play for women. Also, since the athletes getting screwed over in this situation are female, who gives a fuck?

Except for the last line with the swearing, those are the arguments that team Semenya have made in court, and will be making before the ECHR once again in coming weeks. South Africa and other countries aligned with SA on this are using the ideas of queer theory that the affluent West came up with in the ivory tower of academia against the affluent West in the sports arena. And in the court of public opinion, the team that's winning - by a mile - is the one that says so long as it's done by males using "gender identity" claims, cheating in sports is A-OK. If ever there was situation where the phrase "that takes some balls" was apt, this is it.