I am from this background, but whilst my family is economically comfortable And educated (University-equivalent educated since 1920s), elements of these traditions persist.
It is not just a Pakistani thing, but a Subcontinent thing. The dominant culture is quite anti-woman, and up until recently widows often had to die with their husbands as they were unwanted by society (Suttee)
Marrying within the tribe was ok centuries ago, when the weakest would be not make it to adulthood, or not be able to work and marry. But today, marriage within a clan or community causes significant health issues.
To OP: If the lady has said that “cousin marriage” leads to her family having weak bones, perhaps she doesn’t realise the true extent of the genetic issues that are prevalent in her family, there could be many other health issues that will remain undetected.
In my opinion: There are logical (to her) reasons for her denial:
-Yes, the blame will be laid at her feet, her husband is of course not responsible for the dodgy genes and the sickly children 
- Yes, the husband will be encouraged by his parents to divorce her and marry a fresher, younger woman, probably another cousin, fresh from the homeland.
-This would lead to a tough situation for her, as if she (I don't know the background) has no income or job, and her residence status is dependent on marriage, or her husbands job, she will be in major trouble, as she and the girl children, and the sick child, will be sent home so that a new wife can hopefully bear healthy boys for them.
- This fixation on boy children is not a religious or Islamic issue. It is an Indian subcontinent Issue, and comes from the dominant culture of the subculture where girls have no value. You have the same in Indian and BangladeshI and Nepali families, even living in England. You don’t necessarily have this with arab or iranian families, where a different culture is prevalent.
- if she is sent home with or without her chIldren she is f*** because the value of a divorced woman on the Indian subcontient is nill. Worthless. A woman who is labelled for not bearing healthy children is worse than that. She is the scourge of the family and has brought harm to them. She will have no income or will have to beg from relatives. She will have no home of her own unless she has her own income or family money.
-Since she married and went abroad, the family members who remained have prior claim to family property and will stop her trying to get it.
- There is no state schooling in her homeland so if she is sent home, and is from a rural or poor background, her children will have no schooling and no future
Even if she is not sent home, the results of genetic screening would be evidence of dodgy genes, and would mean that her children are not marriageable (even though the entire family share the genepool, so perhaps that is why she doesn’t want the tests done. If there was an option to have the screening done confidentially, without her husband and in-laws finding out, that may be an option, however it brings only negatives for her.
Thank you for trying to understand her. As a child I argued with my family about these topics, we had many scientific and political debates.
I hope her life gets better and the cycle isn’t repeated for her children. This isn’t a cultural but a class and a feminist issue: her socio-economic and resident status here is the problem. If she had solid residence and her own income or state help, enough to eat and send her children to school, keep them warm and give them a roof, I think she would more willing to dare, she sounds powerless at the moment.