You suggested that Oswald needed an alibi in case he was rounded up in a raid. And that explains why his signature appears on that date. That is idiotic for the reasons you apparently can't follow. If person A is arrested at a certain place and time, then the authorities know he was at that place and time because that is when and where they took him into custody. A person can't be two places at once. So Oswald's signature does him no good in the silly scenario that you have proposed. Nor does it make sense if the authorities have the capability to get him released. You don't need an alibi if someone in charge of your incarceration is willing and able to release you. They just arrange for his release. Most likely his is a hoax after the fact.
You suggested that Oswald needed an alibi in case he was rounded up in a raid. And that explains why his signature appears on that date. That is idiotic for the reasons you apparently can't follow. If person A is arrested at a certain place and time, then the authorities know he was at that place and time because that is when and where they took him into custody.
Psssst... Mr "Smith"... If an undercover secret agent ( Let's say CIA) is rounded up with the criminals that he has exposed, the agency sure as hell don't want to compromise that agent by putting him on trial.... They would be prepared to make that agent be far away from the area at the time and therefore he couldn't possibly have been the person who was using the name "Oswald" in Luscome....
PS...This is simply one idea and explanation that the agency might order an agent to go to a place that registered the guests, and sign in as Lee H. Oswald....