It’s my understanding that when the Catholic priest witnesses the marriage in the United States, he is basically serving as the legal representative of the State. The USA does not have one marriage for the “church ceremony” and another marriage for the “civil ceremony”. Therefore any priest, minister, rabbi or other religious person who presides over your ceremony is basically also being the legal person presiding over it. Once you are married he signs the form necessary for the state to recognize your marriage. You do not have to have a second ceremony in front of a state judge or any civil authority.
While I’m sure there is some rogue “minister” or mail order “minister” or something out there who would marry your friend (similar to how Jim Morrison allegedly married a woman in a pagan ceremony but never got a license and thus it was not legal marriage), no organized Church like the Catholic Church is going to get involved in something like that and put themselves at odds with the state.
I don’t know if your friend can find some other jurisdiction outside the US where church marriage and civil marriage are legally two different things, but it seems like she needs to reinstate herself with the US state legal system if she wants to be married here. The US uses the institution of marriage for many legal purposes including taxes, child support, social security benefits to surviving spouse, medical consents, etc. (This is also why some people argue for gay people to be able to get married by the state even if the Church won’t marry them.) As the Who sang, “It’s a legal matter, baby.”