Normally we see sole custody needed to bring the child or a letter from the father giving consent to have the child move to the US. If your lawyer is making something sound like the child is visiting. His visa will be denied. All of this is need at the interview stage in your country. You do not send it now.
What country are we talking about, as some countries single/never married mothers automatically have sole custody.
Also sole custody in most countries does not mean the father is cut off. You can still make sure a fair agreement is made for the child.