How can I get ClockworkMod Recovery to properly use /sdcard on an encrypted Galaxy Nexus?
The important part looks to be this:

It's worth mentioning that I have data encryption turned on.

I don't believe ClockworkMod can handle encrypted devices as of yet, which is problematic for devices that lack true external storage support (e.g. SD cards). The reason it's a problem is that /sdcard on these devices is not a separate partition, it's a FUSE mount point for the /data/media directory. Therefore, when you boot into recovery, it can't access your internal /sdcard directory because it is encrypted.
TeamWin Recovery, as of version 2.2.0, does support decrypting your device when booting into recovery. They actually specifically noted the Galaxy Nexus in the release notes, possibly because they had to implement the decryption differently on non-AOSP devices (e.g. support for encrypted TouchWiz devices is listed in version 2.4.0.0):

What's new in version 2.2.0

Support decrypting an encrypted data partition on Galaxy Nexus (enter password using keyboard)


Realistically, the best bet if you want to encrypt your device would probably be to use TWRP instead of CWM. Otherwise you'll have to continue jumping through some hoops when trying to use your recovery. You can download TWRP from their website.