Question about my daughter getting a UK visa

I'm not sure if I've posted on this forum or not but if you've seen this question before it might have been me and I apoligize.
Me and my daughter are americans. My husband is british. We are planning on moving to the UK asap. My daughter is NOT my husbands biological child. I am wondering if I will have any problems when I go and file for my daughter's visa. I called Woodbridge and they asked me what proof I have that she's my daughter I told them a birth certificate with only my name ( her father's name was never put on the birth certificate) and his name is NOT on her passport. The passport agency told me I didn't have to add his name considering it's not on the birth certificate. Also my husband claimed her on his US taxes last year. He was ordered to pay child support by the state because at one time I was receiving welfare. We were never married and there was never any paternity I just said he's the father and he didn't disagree. We havn't seen him in over 2 1/2 years and actually have no idea where he is. I have no contact info for him and seeing as no child support has been paid I'm sure the state doesn't know where he is either or else they would have gotten him and put him in jail. So with all of this do you think I will have any problem? Have any of you gone through this? And do you think I should add his name on the visa? I don't want to lie but her step-dad has ben her true father for the past 2 1/2 years.

Please continue in your other thread.