In his books, Bell did not mention the make of his .303, so I think it was a standard BSA make Lee-Speed. His primary elephant rifle was a short-barreled light .276 Rigby Mauser. He exclusively used German DWM 173gr rn fmj loads, "it's barrel never polluted with soft noses".Here is the picture from his article, but apparently it shows a later, 1930s, such rifle.

The 6.5x54 Mannlicher-Schoenauer supplied by Fraser he tried on elephant he describes as a short, full stock carbine. He praised Fraser for hollowing out the stock to save weight, but this is a Steyr factory feature on 1903s.
His "meat" rifle he describes as a .256 "Old model Mannlicher by G.Gibbs" he most often used as a single-loader for the lack of chargers. He never used it on elephant because he had only soft-nose cartridges, but it slew vast numbers of "plains game from giraffe and buffalo down to feed himself and his safari of more than 100 native bearers. This Mannlicher must have looked like the one below, also by Gibbs 1899.