I am going to mention something as a "matter of fact" just to give you a perspective on this issue. I recently worked with a different LMS in a experimental environment, seeing if this other LMS could work. I found it interesting that this other LMS was able to boast about its ability to provide great information about students' time in courses, they claimed much more than products like Moodle. Well, I used this product both as the instructor, and as a student (i.e., a student account doing some of my own assignments.) What I discovered what how dramatically incorrect this new LMS was. As a student, I would log in, do a quiz, and then leave (not log out). For my 3 minutes of work, this other LMS would report several hours of being in the course!
So if your administration really wants this kind of information, right or wrong, you might need to consider using a product that gives administration what they want.
I am one who believes that the goal is to achieve course/learning objectives in the shortest amount of time. This then allows the students' brains to learn more. But I do understand where you are coming from, and don't take my comments to be negative about yourself, but rather negative about administrators who think this way.