Is it appropriate to ask a professor to bump up a grade when I suspect it may be at the cutoff?
Consider asking for an appointment to review the final exam.

Preemptively saying "I might be below the cut-off and would appreciate free points" is obnoxious. It's likely the professor has already taken your improvement and other factors into account. Further, asking for additional points for any reason other than a grading mistake is really inappropriate, though it's so common that most professors accept it. 

On the other hand, waiting until the final grades might be too long. Once the professor has made a decision about what grade you deserve, they are much less likely to reverse that decision. 

So, a good middle ground is to ask for an appointment to review the exam. This is a good thing to do in any case; further, the professor will take extra care when assigning your grade, as he will anticipate having to justify it to you. This also does not put you in an awkward position if you actually earned the grade you wanted -- you can just review the exam and thank him for a good course. Note, if you do this, you should make a good-faith review of the exam and an honest assessment of what you deserve; this is not "one weird trick" to get free points.