I think Bankmanager Revived or PersonalAssistant were able to do this. Maybe also Inventory Manager or any other inventory rule addon like Rulebased Inventory.
I'm using DoItAll and FCOItemSaver, AdvancedFilters and FCOCraftFilter to mark and filter my items with a new defined FCOIS marker icon (to bank/from bank e. G.), just filter them to only show these as the house bank opens, and then press the DoItAll keybind to move all currently visible (filtered) items then.
I think Inventory Manager didnt really get an update since those storage coffers got introduced.
When I open them the normal bank options are performed ^^
On the other hand its good, if I wanted to assign an FCOIS marker for each storage coffer, I'd have to make Baertram add even more markings (possibly) which might break the menu completely XD