You could give her a good life - but could you give her the best life she can have?

It sounds like the mother is better at caring for her, and she's more attached to her mother. It may be a "if you love her, let her go" situation - let her life with the parent that'll give her the best life. Ask the wife if you can pay child support, so that way you can take some of the burden off your wife and support your child.