Discussed the lead foil/sheet with the DM, and, based on his experience as a materials engineer, it won't work.
Apparently, lead foil exists, and you can wrap it around things for radiation shielding, but only for fixed, non-moving objects. Flex it back and forth a few times (and it'd be flexing a lot if incorporated into clothing, or a possum pouch) and it quickly falls apart in your hand. The 'lead' aprons used to take X-rays don't actually contain a solid, continuous layer of lead.
And, for a more in-game reason, a lead-lined possum pouch would hide the magic auras of the pouch's contents, but not the aura of the pouch itself. To hide the aura of the pouch, the lead covering would need to go over the pouch and, therefore, would no longer be hidden by the pouch's illusion and be easy to find...
Just to clear things up, it's not a spy- or stealth-themed game - it's a normal game, in the sense that the backstory and the overall threats are the same as you'd find in a hack-and-slash game. Just that we're approaching the problems with guile, strategy and allies/minions/pawns rather than brute force, tactics and personal power.