You didn't mention this fact in your question, but it is very important. And that is, when was your girlfriend's custody-child support order granted? If it was within the last two years, during the time you were with her, then the issue of your prior convictions is res judicata (i.e., already litigated or should have been litigated) and not a "change in circumstances."
Even if your girlfriend's custody order was granted prior to you meeting her, your convictions for "attempted" voyeurism and public indecency, seven years ago, are of marginal if any relevance. Put it this way, I would take on your case. The issue is always best interest of the children, and I'm not reading anything about any harm to your girlfriend's kids.
If your girlfriend doesn't already have a lawyer she needs to get one. You both should listen to the lawyer's advise. The fact that you have been together for 2 years without an incident is relevant. However there is no magic document you can get to show that you are not a threat. Listen to your wife's attorney and be available if you are needed to testify. Why is he only going feet the 15 year old? Does she want to live with her dad? If she does, the court is likely to weigh her wishes strongly.
This answer is provided for general purposes only. If you need legal assistance you should consult with an attorney. Responding to questions DOES NOT create an attorney-client relationship.
Sign up to receive a 5-part series of useful information and advice about child custody law.