Let us refine the idea a bit.
Let us use the bank as a way of doubling the amount of money a single character can hold. If VIP has a four million plat cap, with bank storage, it becomes eight million. And let the doubling effect scale with the plat limits placed on premium and f2p. Possibly make this a VIP perk/low TP cost for premium and f2p(I say low cost since it would work within the existing bank that they already have access to).
With the shared bank, let us place money in there for transfer between characters. Place a limit on how much can be placed in the shared bank, maybe something like 100k plat. This feature could be a low cost TP addition (shared bank is already a VIP perk). Maybe even have different tiers of limits, depending on how much someone is willing to spend for the extra plat space.