Things we know: If someone eats two DF they explode. BB's DF power allows him to heal quickly from all damage. Upon consumption of an
entire DF, the host body is transformed. Partial or incomplete consumption (eating only a piece) of a DF does not give a host DF powers (see CP9 members eating the DF in Impel Down story arc).
Assumptions I'm making: Upon eating a DF, it stays in the hosts stomach and imbues them with power, like a symbiote. The DF explode if they interact because they are incompatible with each other.
The cover is easy to explain away as BB doesn't want the whole world to know how to steal DF powers.
While WB is still alive, BB cuts out WB's stomach containing the Devil Fruit, and transplants it into his own body, giving himself two stomachs, each containing a different DF, gaining power from both, without allowing them to touch each other.
WB's damage to the chest hides any surgery done, and BB heals rapidly so nobody can see that he transplanted an organ into himself.
Counter-argument: Couldn't anyone just transplant an extra stomach into themselves and eat a second DF to gain it's power then?
Response: Yes, but if you have two stomachs, there's no guarantee that the second DF you eat would go into the new empty stomach. You could explode. Also, the DF may need to be 'activated' in a separate host first to prevent conflict with an existing DF.