I've mocked up how you could merge the existing Radial Inventory menu to solve a few of the issues that the new inventory brings. The main advantages from this suggestion is that you'd see at a glance what's in a barrel and you wouldn't be "locked in" to a menu while interacting with the barrels.
The main disadvantage versus the new system is that there are only 8 inventory spots on the barrel, but that seems quite a lot... That's still 792 cannonballs in a single barrel, if each spot can have 99 items.
Closed barrel
When approaching a barrel you'd see the contents on the radial with some transparence, without having to press any buttons. The two numbers per item would quickly show you what amount of items you have (the number on the bottom) and how much items the barrel has (number on top). The numbers would be red if you can't store/take any more of that item.
Opened Barrel
Pressing X would open the barrel and automatically select the first item that you don't have a full inventory for. You'd then tap X to take items from the barrel or tap Y to remove from the barrel. This would allow running on a fort and quickly tapping X to get bananas while being chased by skellies.
You won't be "locked" to the barrel. If you pushed the left stick on the controller you could just walk away.
You could have a Take All by long pressing the X and a Store All by long pressing Y.
Adding to the Barrel from the Inventory
You'd select an empty spot on the barrel and press X and it would show you your inventory.
Selecting Item on Inventory
Pressing Y would store from the inventory into the barrel. Pressing X would show more pages of inventory.
You'd only see items that you can store on the barrel type. So if you're on the ship and open a cannonball barrel it would only show you cannonballs on your inventory.
To back out you'd press B, but most of the time you'd just walk away when done with it.