I don't know why a Remington Model 8 or 81 in .35 Remington would not work with any SAAMI-spec cartridge in that caliber.
Just FWIW, I have been doing some more research on the Frank Hamer rifle(s) and I am totally confused and mixed up. As far as I can determine, I was wrong on the caliber as most of the POE-modified rifles were in .35 Remington, though .30 Remington was offered as an option. It appears that he did have such a rifle, which his son showed to the press, but it was a Model 81, so it could not have been used in 1934. Whether he had another POE rifle earlier or whether he used a standard Model 8 is a question lost in the mists of time.
Also I was not entirely accurate in calling one rifle a B.A.R. - it was a Colt Monitor, a commercial version of the military B.A.R. The felons did have B.A.R.'s, stolen from a National Guard armory.
One thing seems certain. In spite of the TV version of history, there were no Thompson SMGs at the famous ambush. But they do look "cool" in the TV show.