The LM2917 is essentially an integrator. If each pulse is the same intensity and duration, then the output is proportional to frequency.
Will all rain drops, independent of size, make the same level of noise? What about sleet, snow, and hail? How are you going to resolve the situation when two or more drops hit the CD platter at roughly the same time?
I doubt any of us can say with certainty whether it will work or not, but there are obvious issues to consider.
You might consider as an alternative approach analysis of the sound. As an example, check out the freeware program called SoundRuler. That program is designed for analysis of bird calls. You can feed it a sound, identify a single call, and then the program will find and identify how many similar calls are in the analysis section. In your case, of course, a "call" is the sound of a single drop. Repeat sampling at, say, 4 sec intervals and you have your drop frequency. Further analysis of the sound may also help is assessing the size distribution too, not just the average volume over a period of time. There are other more sophisticated programs for sound analysis, but SoundRuler, while limited, is simple and will give you a feeling for whether that approach might work.
Sounds like a fun project.
John