One thing you can try is to switch your search date to a few few days earlier, where BA does not have its own award available. The site should then gives you the option to search on partners, and you can use the "next 7 days" link to navigate back to your intended departure date and book from there.
As colmc said, this depends on the partner. BA.com doesn't support some partners, such as AS, and you also need award availability in order to book the flight.