What incentive does a professor have to allow a student to enroll in a course that is already full?
I disagree with your proposed strategy.  The marginal added workload of one additional student is usually negligible, and the average grade for the term is not usually something that reflects well or poorly on a professor to a significant degree.  These aren't usually the main issues.

I would expect the professor to make the decision by trying to balance the benefits to you of taking the course, versus the negative effects on the other students from having the course be more crowded.  Your case will be more compelling if you can show that:


Not being able to take the course would represent a major hardship for you.  Something on the level of delaying your graduation.  If the only effect is that taking it later is somewhat less convenient, means a fuller schedule, means getting up at 8am, etc, that won't be compelling.
The fact that you couldn't register for the course normally is due to circumstances beyond your control, and not a result of inadequate planning on your part.
You are adequately prepared for the course, and are reasonably likely to pass it.  Being a "good student" overall means something here, but more to the point is showing that you have done reasonably well in all prerequisite courses.  If your preparation is marginal, the professor might think there's not much point in giving you permission to take a class you would be likely to fail.


There isn't much you can do about the negative effects from crowding; let the professor evaluate that.  Don't make suggestions like "I could skip all the lectures so that there will still be enough seats".  That tends to counteract #3 above.

Definitely do not attempt to bribe the professor with money, favors, etc, in exchange for being allowed to take the class.  (This should go without saying, but I have known it to happen.)