Honesty I never used the bank from any BL game all that much. Even sharing weps between characters had limited use from my POV.

When I get a new weapon that outclasses one that I have, I sell the old one. I like to keep a wide variety of weapons types, sometimes trying to ensure I have decent elements among all weapons types (for example: I'll keep an underpowered fire elemental pistol even if I have another pistol that does more damage, just to make sure I can inflict burn damage with pistol ammo if I run out of ammo for my Legendary Fire Elemental SMG). So my priority w/r/t upgrades is to max out my backpack slots.

Bank slots have always been lowest priority.