Ive been playing BL3 since launch and played the previous incarnations, but its my first post on the forums and dont know another way to pass on a suggestion, or if it has been made already. so far what ive read has variants on this, but not exactly this.
Go ahead with the expanded slots in the bank, that will work fine for the shared items.
i would suggest to the developers that the individual displays in each characters room, should rotate similar to vending machines, so you could use those to store a number of weapons in each type that you use on that character, and then the shared bank is for trading between characters and excess storage.
say each of the panels was made into a 4 sided rectangle, and it rotated on a horizontal axis using C/V or some other keys to cycle through the weapons.
of course there is the consideration of the logistics of programming this, but it is a suggestion to use an already existing feature to resolve an issue