The spellings I have found include Beaushyn, Beaushin, Bewshine, Beaushens, Beawshen, Beyssyn, Beschyn, Bewshyn, Besschin, Beawschyn, Beauchier, Bewshin, Bewschen, Beauchin, Bessyn, Besschin, Beuashin, and are connected with the manor of Beauchinhays, Bessines-hay, Bessynishegh, Beawshenysheyis, Beawshene, Besschinneshegh, in Whitchurch Canonicorum.
It doesn't seem to be connected with Beauchamp (which I actually have in a completely different branch).