Thanks for the input, Wendy.
I should have updated this thread about a week ago.
I've been using the batteries that Cochlear sent me and honestly I haven't seen a big difference between them and the Duracell batteries I usually buy at my local Walmart. The sound from my processor is just as good, they last just as long, etc.
I think the low battery beeping problem I was experiencing was related to my BP100 "reprogramming" itself. One day while driving to work with my iPod on (I have an adapter for my car) I realized that my BP100 was beeping in response to certain musical notes in certain songs. Over time, it's stopped doing this.
I remember that when I first got my BP100 it used to beep in accompaniment with the seat belt reminder beeps in my car. After a few days that stopped.
The BP100 will beep in response to "strange" noises it encounters (like the sound of the beeps on your microwave, etc) until it realizes that these are "normal" sounds in your world.
In the past week I have had absolutely no false low-battery warning beeps from my BP100.
I think the program is solved - or at least, I'm keeping my fingers crossed.