Most everything above works.
It would be possible for a non-student to enter a university library and look around the bookshelves. At least during regular hours where I live (US).
But, my university students do need to have a current university ID to enter the library after regular hours (for safely reasons, as the library is sometimes open all night). This is like a key card, so is only supposed to be active for current term students. You might need your students to access the dimly lit halls late at night, when most of the library is empty (but not all of it, believe it or not).
And, many students just do online research, not physically entering the library, and you need a login for that (as a current student) to "check out" material online.
For your needs, there are also a few other options. Do they have to check the materials out? I think I remember you said they just need to scan them (perhaps take pics with an iPad? - this can be as good or better than most conventional "scanners" can be, although probably not for old faded documents, or with bad lighting). Could they do that in a corner of the library, with books hidden in some rarely-touched shelves? (some of the shelves even have dust on them now, because so many research online instead).
However, you might consider that most special items, old or valuable books, items out of circulation, etc are kept in a contained place in a library (and sometimes only open during certain hours).
As a professor, I can also have a set of items on "reserve" under my name, as someone else mentioned above. So, if the people were students, they could go to the library and view these items just by asking for professor xyz's reserves (but often they are not checked out, just made available for viewing in the library- but it is possible that the professor could allow them to be checked our for a very short time too).
On enrolling them as students, this is usually done online now, as mentioned above. I should add, however, that most universities still help out students who need help, but students often have to wait in a long line at the admissions office (or registrar's) for such general help.
Also, if it is a class that has prerequisites, and they want to bypass them, or need to be "cleared" to register, they often will meet with their department academic advisor, and sometimes the professor has to approve it as well (even in early registration). I have had to do this teaching at two different universities before, approving students, or enrolling them during a meeting when I was their academic advisor). Some departments make students get cleared first to know they are on track for their major (then they can go finish registration after that). So perhaps your professor has that access and can add them then, or give them clearance.
Also, as mentioned above, they could be added after start of class and need the professor signature to be added.
But what about an independent study course? That could be one credit, working directly with the professor, and give them full access to the library? I have seen this done too, but it usually needs approval by the department chair (perhaps that could be arranged in your story?) This could be explained as a student assisting the professor in research as the student also learns as they work.
Lots of options