Not a major thing, but little in DDO makes me more upset than loosing an auction simply because I'm not on the character with whom I'm bidding.

So my suggestion is to simply send mail to the character you're playing at the time, as well as to the character who bid. The mail to the bidding character would include the money, while the mail to the other character would merely read "You have been outbid on an auction for an <item> being bid on by <character>"

The only drawback I can foresee is that it might cause a bidding war eventually, but the circumstances now don't seem to indicate that that will happen much.