Of the cartridges you selected the recoil from softest to hardest all else being equal would be:
.270
.308
.30-06
7mm Rem Mag
For the game you plan on hunting any will do the job. But having all but the 7 mag I've used the .270 a lot over the years for the game you mentioned it performed flawlessly with 130 gr bullets taking game from in my lap to well out past the 400 yard mark.
But like some of the other posters have said the 7-08 is a remarkable little cartridge that is relatively mild recoiling and hard hitting.