I think because of the -beth part of her name you would get away with it as long as it was Bee not Bea.
It is only a nn so it wouldn't be ridiculous at all.
other nn I found are Bess, Bessie, Bessy, Beta, Beth, Betsey, Bette, Betti, Bettina, Bettine, Betty, Buffy, Eliza, Elli, Els, Elyza, Elzira, Ilsa, Ilse, Libby, Lilibet, Lilibeth, Lise, Liz, Liza, Lizbet, Lizbeth, Lyza, Sissy, Telsa, Lizzie; Liesl, Liezel, Lizette.
So compared to them Bee is positively normal