Account Bound "things to do"
I can understand that you can find going to your bank multiple times a bit boring. Nevertheless, don’t you think it wouldn’t be very role-play to have a bag shared between two characters ? A lot of things in this game are already designed to be convenient without being particularly role-play (collections, free armor repair, etc.), that’s why I wouldn’t support your idea. Collecting tools could become part of the collections, however : have 150 axes on you while travelling already isn’t role-play.
To transfer items, map travel and use the WvW bank. You can also use the other services (merchant, crafter, repair) if you want. When you relog that character, you will be in the same PvE map spot that you left.