First, put 10 similar distressed assets (such as a series of collateralized debt obligations) up for auction. At the close of the auction, the Treasury pays the winning bids for nine of these properties. The 10th property (chosen randomly) gets sold to the winning bidder...
You'd have to worry about auction-rigging, in which leading bidders collude to bid low, but there are ways to discourage that—say, with a sealed-bid auction in which the winner pays not his own bid but the fourth highest.
I thought about this only briefly, but it seems to me that there are some skewed motivations here when the banks are both bidders and sellers.
Suppose bank A has three near-identical assets in the auction "worth" about $1M. What would the bank bid for its own assets?
Suppose it bids $1M apiece. Then the EV to the bank is bounded below by zero. Worst case is that two of the three assets are bought by the government for $1M each, while the third is bought by the bank itself for $1M. The bank gets rid of $2M in distressed assets for $2M in cash.
Now suppose it bids $2M apiece. The EV to the bank is now bounded below at +$2M! Worst case again is that the bank has to buy one security at its inflated bid but gets the government to pick up the other two. It sells $2M in distressed assets for $4M in cash.
As long as there are 2 or more roughly-equivalent securities in the bundle being auctioned, a single player acting alone (as both buyer and seller) can inflate prices without risk. Sealed-bid auctions mute this effect but still don't provide an incentive for a bank sitting on both sides of the auction to arrive at a fair price.