That's not how the heat patches I've used work. They're normally full of fine iron powder, the reaction is limited by available oxygen. Heat is generated as iron rusting (combustion) is exothermic.
I've made decent desiccating devices based on calcium chloride, the common road salt variety. Just seal it in a container and don't let the stuff you're drying actually touch it.
I don't think it'll work well for your case, though. What might work is to get some epsom salt, MgSO4*7H2O, pour it in a nonstick pan, put it in your oven set to 550F or higher for an hour or so, and then bust up the resultant salt cake before it gets a chance to cool or grab any moisture from the atmosphere. Bust it up a bit, get it in a very airtight container, bust it up more by shaking the container, cool the salt in the container, and then seal the prefrozen rat brain in the airtight container such that the brain is isolated but atmosphere is shared.