Matt, first off, I'm sorry that you're having problems with your gear, but there's two comments in this thread that I'd like to respond to.
FWIW, I noticed that even the metal snaps on the carrying case are rusting.
To me, this is the key comment so far in the whole thread as far as diagnosis. If you're seeing corrosion on other pieces, then the culprit probably isn't anything specifically related to the HDP2. You need to concentrate on the environment...you must have some corrosive elements in the environment either in your bag or near where you commonly store your gear and bag.
I really think Tascam needs to step up on this.
It's really pretty hard to know if this is Tascam's issue to resolve. They use high quality Neutrik parts just like other manufacturers and since this doesn't seem to happen on other units, you can't really say that it's an inherent design or materials selection issue.
Corrosion of metal components can happen anywhere the physics supports it happening. For corrosion to occur, you need an anode, a cathode and a transmission medium. What happens is that the metal on the anode is removed and tranmitted to the cathode. The driving force for the metallic transmission is an electrical current...which often is naturally occurring, but not always.
In the case of the P2, it's fairly obvious that the transmission medium is humidity. I think that, even with desiccant packs, it's possible that there can still be levels of humidity in the air that will be high enough to support corrosion. In the extreme, any humidity at all could be too much.
The metallic components on your P2 could be acting as anode and/or cathode. Any two dissimilar metals can act as the anode and cathode...and thus they can cause a natural electrical potential to occur which will cause corrosion (in materials science, that's called galvanic corrosion). However, I don't think that's what's happening here because if that were true, it would happen alot more often on alot of gear and we'd be talking about it alot more. Also, for galvanic corrosion, the corrosion areas are at the interface/contact areas between the two metals and I think the pictures you're showing show that the corrosion is general surface corrosion.
It's possible that, if you aren't removing the batteries from the P2, that the battery case inside your P2 might not be grounded well and that's what's causing the electrical potential to occur that is the driving force behind the corrosive action. I might suggest taking a peek inside to see if there are some loose wires...anything that looks like it's causing abnormal grounding situation.
The strange thing is that you say that pieces outside of the P2 are also corroding. If the carrying case isn't electrically connected to the P2 somehow, then in my mind that would rule out any battery or grounding related issues (although I suppose it's possible that when it's sitting in your bag, there could be incidental metals in contact causing electrical continuity). However, if you can rule out electrical continuity between the case and the P2, then in my mind that would completely rule out that the P2 has anything to do with what's happening. I would focus on the transmission medium in that case...humidity and perhaps whatever corrosion inducing products might be suspended in the air.
Personally, I wouldn't put my P2 in a plastic bag with desiccant packs, at minimum unless I did some more research. Some people assume desiccant packs are good for our gear, but do you know for fact that's true? I don't. You might be creating an even MORE problems for your gear. How can you be sure that, when you seal the bag, you aren't sealing in an even higher humidity environment. Are you sure that the desiccant packs are designed for the environment that will be encased inside the bag? What happens inside the bag if you happen to forget to remove a battery and it starts to corrode? Is the gas or acid now sealed inside the bag gonna screw up the circuit board or chips? What if the bag somehow gets exposed to the sunlight or extreme temperature changes and starts to sweat internally? Bottom line for me is that it seems to me that if desiccant packs were that good for electronics, then they'd be included with the OEM packages when we get the gear new. My electronics NEVER comes with desiccant packs inside the plastic bags that the gear comes in, so I'm not using desiccant.
Unfortunately, I don't have any pat answers for you, but hopefully I've given you enough information to help a little with diagnosis.