Well, Bedivere collects artifacts himself.
He would have already pursued this path should he believe it work for him.
It makes no sense that, such a stone randomly appear as a plot hook given Bedivere's history.
Thus to negate this falsehood...
Logically since he hasn't quite ruled the world yet, he finds no interest in the stone.
Most likely because he does not believe it to be able to work for him.

After all, he's lived far longer than Martin and has quite sometime to do research on mythical objects a collection of which he houses at Camelot's trophy room.

If Bedivere did after all use the stone, it would make him incredibly OP and the RP would be over in a matter of minutes as Bedivere kills/converts/enslaves every last human in the course of about 40 days and 40 nights, throwing a new dark age of vampires to effectively end the power struggle as there is nothing one can do to stop him as his one major weakness was rid of by the stone.

Also, given that it is from Martin, who is a human, Bedivere's elitism would scoff at using a weapon given by a human. There is no honor in using the boon given by the humans to kill the humans. You've effectively become their puppet, and Bedivere is not a puppet by far... (Well he is, for the Pale Lady.)