WADR, I believe you are confusing two different cases.
An agunah is allowed to marry based on the testimony of one witness. However, if she does so, she is required to perform an investigation and if her first husband shows up, she’s out of luck in that she loses both husbands.
If she marries on the basis of two witnesses and with the permission of Bais Din, however, she is afforded the protection that if her husband returns, she does not have to leave him (the first husband). In this case, the marriage is sanctioned by Bais Din and is perfectly permitted. And yet, her kids (with the second husband) end up as mamzeirim anyway.
The Wolf
Nope.
The fact that it was sactioned by BD does not allow her to remain married to the second husband!
The “Assur L’Baal Ul’Boel is waived “Mipnei Sheasah B’Rishus”, but that is NOT an issur Torah, since there was an Ones (see Rashi Yevamos 87B). The children from the second relationship are still Mamzerim.
The first case (where there is only one ayd) is to be machmer on her at the end so she should check well at first (See Yevamos 87B). Also see there regarding relying on BD.