Thanks for the replies, guys!
After reading your replies and trying the Texture Packer directly (and not using the texture from the Export feature), I realize my issue was a very simple fix, since I can just pack the images first and refer to the resulting image.
We are using the cocos2d runtime. We're experiencing issues when there are about 20 or so units on the screen (which are composed of 8-22 bones/images depending on the unit) along with various other sprites/particle effects on an iPhone5. I don't believe the framerate drop is due to the number of spritesheet files we are using, but we shall see.