When you go to the Bank Deposit Box, there are Buttons. Store Inventory, Store Equipped Gear and Store Looting Bag.
Would like to expand on the "Store Looting Bag" to include some of the other bags in the Game:
Fishing Barrel
Log Basket
Plank Sack
Coal Bag
Seed Box
Herb Pouch
Gem Bag
etc.
And maybe we could put a toggle on it, similar to the new Slayer Block task, where you can go to that tab and turn off or on the bags you would like to have deposited with that button.