I like high stamina but unless there's a blowout on your weekly schedule, the minute management problem will exist.
One way around it is to play the player with high stamina once as a starter and twice as a backup during a given week. Or you could play him 48 minutes as a starter one game and as a backup in another. Since he has high stamina, his production won't dip much by playing him 48 minutes in a single game.