First off all, a mamzer is only the offspring of an illicit affair with a jewish men and a married jewish woman. If the man is a non-kosher convert (meaning not-jewish) then halachically he is not the father and the child is not a mamzer
If the mother is a non-kosher convert, Then the child is not jewish so they are not a mamzer either.
Most gerius is not a joke, Most people who convert genually want to observe the Torah and mitzvoh, there are some who convert for marriage reasons (Ivanka Trump I belive) , You can prevent some of that, but not always.
Except for the Vaad of Queens , there is no standard conversion. Its basically whatever the Rabbis decides and its also clear that just because someone is a Charedi Rabbi doesnt mean he is Lesham Shamayim either.
Rav Moshe actually wrote about Rabbis who were pressured to do converstions (Im guessing because of marriage reasons) and generally held they were valid.
I do know of cases where a convert married a Cohen. The convert had converted conservative. At some point they became BT’s and they had to convert Orthodox. The couples were allowed to remain married and any offspring was not considered a Mamzer, although they are not Cohanim and there is no Valid Ketubah.