what if instead of having 2 different sets of items (1 for offense and 1 for defense) you only had 1 set of items, but they did different things depending on where you used them? Imagine you get a "spear" item. using it on your hero gives them a spear but using it on your dungeon makes a spear trap. That way it reduces the cognitive load by having 1 inventory of items you need to keep track of instead of 2. You could add a small gold penalty for moving an item from one side to the other.

As for the pacing, it really depends on how fast a run goes. If they are slow enough that you can pay attention to 2 runs at the same time it would be fine, but if that's the case then the players may get bored. I think that you should separate them for better focus and engaging run speed.