I was thinking the Treasure Bag size would be adjustable in the Skill Tree. Those who want to carry lots of items can sink points in Treasure Bag size, and those who don't can distribute them elsewhere.
To expand on this idea, I was thinking you could save Skill Tree customizations and name them for teams you consistently use or certain dungeons you expect to trawl through. Some skills would be linked to an individual teammate (such as Type Bulldozer or PP Saver), so there would be incentives to allow specialization by cheap distribution of points. Unused points would contribute to increasing the amount of money you find in dungeons.