I do the alternating thing. I only do 1 heavy set after my warm ups so I just switch which hand is supinated each workout. It feels really weird the first time you switch grips but after a couple workouts it will feel normal. It's working fine for me.
I don't know why you would use straps. I always thought one of the best things about the deadlift is that it increases your grip strength. If your grip isn't strong enough, why not just lower your weight until your grip catches up, instead of using straps and making the imbalance worse?