My university requires that I take attendance. How should I treat students who arrive late to class?
This is an answer to a previous version of the question, where it was not clear that recording attendance was mandatory at the asker’s university. Therefore my answer does not cover the specific situation of the OP, but may still help others in a similar situation.

0)  Allow all students in class and do not monitor attendance.

They are adults and they come to learn. As long as they do not disrupt the class they are free to come and go as they please. I do not see which purpose the monitoring of attendance serves. It sends the wrong signal and focuses students on signing the register sheet rather than on the objective of the class.

I do, however, start each class by thanking students who attend and come in time. Latecomers will miss this part.