This thread is sad.
Clearly the dc of the OP wants to help a child by fostering but the OP. is very aware that just the association of the child young person being a Foster child ( the name is the trigger here as from what I read from op updates she would not feel the same if it was an adopted child or biological child. Yet all children can mis behave steal). Back to point, it's not that it's a child that needs help but the deep routed trauma the OP associates with it being a Foster child.
Very similar to say trauma triggers some people have with a persons names, smell or even sounds.
The OP clearly cares about her own child and don't want to actually cause friction but is acutely aware this will pose an issue for them to Foster. Which to be honest I believe it will.
There is a post further back from a person who to be honest gave the best advice from what I could see.
I can also see that the OP said she has had therapy and still feels this way. I have no idea if that is ongoing or was of any help.
However my advice would be to maybe have some more therapy if she don't want to prevent her dc from being able to Foster. And if this is a boundary she can not work around. ( therapy don't cure don't make everything better rather helps to process the feelings and learn your own coping strategies and boundaries).
Then unfortunately it will be a case of if dc does Foster that family will be fractured.
This is not a case of right / wrong but rather the complexity of broken families and the care system. It shows how and why there is so many children in care.
OP - I wish you well and hope that you and your DC find a solution together.
I am sorry I do not have any advice to give. Or can answer what you should do.
I don't think either of you are being unreasonable but rather it is a complex thing that requires support and guidance from trained professionals ( social care and therapist).
My post was long sorry, take care.