I think that there's a reason for the current system, although I don't know what it is.
Yep, of course there's a reason. It's got to process all those transactions in order to send payment to the players who just sold items, to send items to the players who just bought them, and to return unsold items to the sellers. Ever notice the slight hang in the game on each 5-minute marker? That's the auction house clearing. Allowing players to pay a "rush delivery" fee would result in the auction house being forced to search for those "special" transactions and process them on-demand.
For those who don't understand that, let me just say this:
OP's idea == more database strain == more lag == bad