Whitman used a Remington 700 in 6mm caliber.
I am certain that our book said he used a .264 Winchester model 70. The reason I remember so well is I was just into handloading and knew the .264 was a fairly uncommon cartridge. It used the same bullets as 6.5 Swedish Mauser but cases would be hard to come by.
Of course the book could be wrong as they often are. I just did a short search on Google and noticed a Remington model 700 in 6mm Remington was being sold as Whitman’s rifle. That would be a good choice too.
I still am not totally convinced that it was not the .264 mag but you are probably right.