Is your daughter mature? Mine is and was from 11. She knows right from wrong but she also knows love.
Taking 12 months away from seeing her Mother, her Mother seeing her daughter is that something you are prepared to do?
Your "wife" is showing remorse you can express to your daughter "why" she commited forgery I am sure you know what the reason was.. To live? To survive? Feeling down about her life, wanting more for her child, thinking they could have a better life, she could and give more to her child?
This is what you need to establish and discuss with your daughter... Or that she was lost , for that moment.
People make mistakes throughout there life, all their life really how many have you made? Some that were illegal ? Drinking / driving, texting, something? It happens...
Discuss this with your daughter, her Mother is her Mother you already know the answer as you are defending her, here in this thread.. It was an error..
Don't look at the environment rather the reason, outcome, and love.