My personal and professional experience of this is overwhelmingly similar to Neo sadly.
OP, it can work, but it'll take a lot of hard work and determination. I don't know exactly what the SW is proposing or whichever LA you're under but on the whole kinship care is cheaper than foster care / adoption, hence why this government likes to push it regardless of outcomes.
Every child that comes into the system has been through trauma in some way, but 4 months is probably young enough for you to be able to give the kid a shot at a normal life. But no one can predict the future, especially if Mum will come back in and out of the kid's life forever more.
What about her other kids? Most adoptees nowadays have some contact with birth family, especially siblings. At some point this child or one of her other children will ask why you took this baby and not the others. Can you answer that honestly? I'm not criticising you at all, but I've known this kind of situation to cause incredibly deep resentment, with far reaching consequences. It's not a reason not to do it but it is worth considering.
Your own children are very young and they need you to be present for them 100%. They need a stable home they feel safe in and fostering doesn't always provide that. My parents fostered children from when I was young. While it shaped my personality, life and career, and I had some life-changing, positive experiences, when it was bad it was terrible. I don't know if I could put my kids through what I went through.