Honestly, signing a message doesn't prove anything except that they know someone who has 43 BTC and they're willing to pay the transaction fee to move it to a separate wallet and sign on Betcoin's behalf. It doesn't prove that Betcoin owns the wallet.

However, their refusal to sign tells me that the funds are not segregated (if they exist at all). What is the harm in signing a message when it doesn't really show anything? Nothing, unless 1) you can't actually come up with 43 BTC, even if it doesn't belong to you or 2) they don't want to show an address that had its balance go from zero to 43 BTC a couple days ago because that would show that the funds hadn't been segregated all along.

The only way that Betcoin can "prove" anything is by shipping 43 BTC to the BBJ winner. To be honest, all of this delay is probably because they're having issues with the code that will let them pre-select the BBJ winner so that they can redistribute the funds among themselves at a later date while not having technically "stolen" anything.