Popa, I don’t agree with the question. We’re never going to have reliable information on a beis din’s success rate in converting gerim who remain frum, and even if we did, we wouldn’t know if it’s due to their low standards, versus some other factor (like luck, or how the local community treats gerim). I do think, though, that gerim should stick with the major batei din (those on the RCA or rabbanut lists for example). Even so, gerim shouldn’t be punished for going to the “wrong” beis din — each ger should be considered on a case by case basis (discretely by rabbis — not on public forums by anonymous commenters.)
Health, review the Shulchan Aruch. The acceptance of mitzvah does not require, and never has, that the ger have 100% accurate knowledge about what all the mitzvos are and how exactly you follow them. If that were the case then we might as well give male gerim semicha at the same time we convert them! Ignorance about the current interpretations of one or more mitzvos does not indicate a lack of kabalas mitzvos.
Furthermore, headcoverings are a special case, because many generations of pious Ashkenazim didn’t cover there hair, and this has continued to this day in some MO communities. So if a woman converts in one of those communities and right afterward marries and doesn’t cover her hair, it probably shouldn’t indicate she didn’t accept the mitzvos (unless, perhaps, the beis din discussed the issue with her and specifically told her it’s a mitzvah).
Now if she’s not covering her hair immediately after conversion in a community where all the women do cover their hair, that’s different, but because of the special history and status of this mitzvah, I’m hesitant to say it would necessarily indicate a lack of kabalas mitzvos. I’m also hesitant because I think it’s inappropriate for anonymous commenters to give halachic opinions on actual cases (such as the case discussed in this thread)! Determining whether a conversion was valid (and thus whether a gerus l’chumra is necessary for her or her children) is a very sensitive issue that should be handled discretely and privately (like the issue of mamzerim).
For example, Health, if one day your daughter wants to marry this woman’s son, then you can ask your Rov to discretely do some research and figure out whether the son is going to need to do a gerus l’chumra. Or if you’re approached about the possibility of a shidduch with him, then you can have the research done and say no if your Rov concludes the gerus wasn’t valid. There’s no reason to discuss this publicly.