I had a .50 AE in the Desert Eagle, man was that a fun pistol. The 300 grain Gold Dot was designed for the AE IIRC.
I currently have the BFR in .500 S&W, with no brake, MAN can those 500grain SP's kick!....but boy is it fun
Start with the AE & make your decision on the level of recoil. It will dispatch a whitetail with no problems & the gun is heavy enough to mitigate the recoil some. Stepping up will provide more POP.... on both ends!
Congrats on a fine pistol.