Government agencies are expressly prohibited from preferring one religion over another, or one church over another. This fundamental principle of the U.S. Constitution- which can often times be lost to some government bureaucrats and they develop a mindset that everything must be "approved" or sanctioned by the government.
In most places, employees of the government agencies are well aware of the First Amendment, and will not cause any problems for you as a Universal Life Church minister.
In some smaller locations, however, ULC ministers have run into opposition when they ask a bureaucrat a specific question such as "Are ULC ministers permitted to officiate marriage?" The church which has ordained you has no bearing on the question, and such questions can confuse someone who is not well-trained. They may not know the answer, but be unwilling to acknowledge their ignorance, in which case, you can face hostility.
If you wish - you can confirm that you can perform a wedding in a certain jurisdiction, it is suggested that you ask simply;
"What is required for a minister to officiate marriages?"
If they ask you what church you belong to, you can simply reply, "Is that important?"
This gentle reminder to them that it is not important, and does not bring you into conflict when trying to register.
You can also ask them if the requirements are different for "non-resident ministers" or ministers who do not live in the state where the marriage is to be performed.