Corrected: BTC not DOGE. This whole situation is confusing.

They don't have any doge to send. They don't even have the BTC, because the BTC is not in an address they control.
The BTC didn't end up in an unknown address. The website can use the DOGE hex private-key to spend the coins in the related BTC address, exactly like HCP described url=https://bitcointalk.org/index.php?topic=5263379.msg54860249#msg54860249]here[/url].