Its seems you like the .303 but want a .308" projectile due to variety of more bullets available, I would look at.....

.300 Flanged (Rimmed .300H&H)
.30R Blaser
.30/40 Krag..
.307 Win.

looking at the above they all will do the job but I would stick with the .303 British cartridge,

Cheers
Steve