Well according to you all if nobody gives a crap what I think when I mean well because I am not the mother of this child, (even if I support his father's right to see him, even if I treat this child very well when he is in my home and even if the child has hugged me and told me he loves me repeatedly)then what do you care what I'm going to think when I decide since I'm not the mother of this child and have no rights and NO ONE CARES if I mean well that I am going to stop worrying about it and stop caring because it doesn't do any good? And what if he and I stop trying to fight the exwife just to see his child and just move on with our own lives and focus on the child hat we have decided to have together? Oh! But that's not right either. I'm sure all these negative comments are coming from jealous exwives yourselves as it seems to me that any woman who is NOT jealous of her exhusband moving on would be very HAPPY that her child has yet another adult in his life who cares about him and his well being and not fight so much for such a person NOT to see him.
Stop obsessing about his ex. You will have no more legal rights to her child than you have now, so being gf or fiancee or wife makes zero difference.
If you and bf have nothing to hide, then being "investigated" shouldn't be a big problem for you. Let bf and his ex deal with their communication and their child, and stay out of their way. You chose a man that has a previous child, therefore you are obligated to treat that child just as you would your own, but without the legal rights. That's what you agree to when you decide to be a "stepparent".
If it's so difficult for you to live with the situation that comes with your bf having an ex and a child, you shouldn't have let him move into YOUR home, and absolutely shouldn't marry and have a child with him. You knew the situation, so complaining about it now is too late.