My question to you is: what's your question? The title and body seem to be two different situations...
Ignoring the body of your question and simply answering the title, the answer is maybe. Typically, a parent won't lose custody simply because someone has to register. But, depending on what the charge was (for example, CSC 1st against a child) it is very possible they could lose custody, especially if the offender lives with the girlfriend.
It would be best to consult with a lawyer.
There are many factors involved in changing custody, and essentially, she could lose custody because she is dating a registered sex offender. There are procedures and best interest factors involved. It is complicated.
If she is sued for custody, she should not ignore any paperwork she is sent and she should consult an attorney immediately.
There is no cut and dry answer for this question.
***Answering legal questions is a complex matter and cannot be done without further research. My answer here is based solely on the information given in the question and should be taken as informational only. This is not legal advice. My answer here does not indicate that I represent you now, or that I will represent you at a later time. I take clients on a case-by-case basis, and only after initial consultations. ***