I've noticed that the average automobile can carry both a rifle and a shotgun without exceeding the GVW. So, me, I'd take both along on the trip.
But I'd definitely use slugs. No way on this earth a clean hit would hurt enough edible meat to cost a sandwich's worth. And, anyway, I'd bet a bad hit with an '06 would make a bigger mess than a shotgun slug.
For the .357, most folks use a lead bullet with a large meplat; around 158-grain, maybe heavier.