Can anyone recommend some type of case or wrapping in which I can store my compass in my pack to shield it from all the other batteries, magnets, metals, etc. rattling around in my pack?

I'd venture to guess that one of those anti-static bags they use for computer parts would work well. Those bags should form a pretty good Faraday cage around the compass and shield it from any outside electric and/or magnetic fields.

THe bags that computer components come in are ANTI STATIC. There is very little you can do to protect you compass from magnetic fields.

From what I know the big thing is to keep it away from magnets such as on speakers. Having it stored near iron-containing materials isn't desirable, but I don't think it will damage it much.

Ironically batteries don't seem to have all that much influence on compass. They are "chemical packs" and only influence the compass because of the metal in the container/electrodes.

Yes, even a faraday cage is protection from the electric field, not the magnetic field. Most modern compasses use either rare earth magnets or some variation of ALNIICO and are fairly unlikely to be demagnetized by common objects at normal temperatures.
Unless you pass by an MRI or a particle accelerator.
There is mu-metal, but I don't see the need. Just package it with normal things, avoid things with very strong magnets (some speakers do in fact have strong magnets, but not headphones)
Brunton Pocket transists have the ability to demagnetize some compasses (ask me how I know).
And some batteries do have magnetic componants and a magnetic field, which while it will NOT demagnetize your compass it can deflect if from the correct reading if it's near when you take a reading.