I have hunted the rez a lot.As I understand,if the Indian land(that's what my native friends call themselves,so I say it is correct,and proper),borders "white man's land",(for lack of a better term),your SD license is valid.If the land is landlocked by Indian land,you need a tribal license.I suggest getting one anyway.It's cheap insurance and a goodwill gesture.
As far as how a white man is treated,the last time I recall any issues was in about 1973,when the wounds were still deep.I love the Lakota people and respect their culture.If you are the type who can't leave your ego at the border,you may well have issues.If,on the other hand,you are respectful and understand that you are operating in THEIR world,you are in for a great time.Don't look them square in the eye and exhibit any dominant body language.Ask a lot of questions about their lifestyle and culture.Don't debate,just listen and enjoy.Help them cut some wood for the elders,or some similar act of kindness. Most of all,spend your money!They like cash,same as you and I.
There is an "Indian" Warden,who has jurisdiction over natives.They can detain you for the "white man's" warden.But cannot cite you.Same goes vice-versa.Don't expect, or assume, that any law enforcement officer will come to your aid. As they explain to me,don't do anything you don't have the horsepower to handle on your own.The last interaction I had with a BIA cop was when I was sitting in my millionaire rancher buddie's 1950 Chevy which hadn't been licensed since the 60's, on a paved Highway between Pine Ridge and Wounded Knee,spotting Whitetails.I glanced in the rear view and saw cherries (gulp),I told him what I was doing.He ripped the binocs out of my hand and started looking, too.Then all he wanted to know is who owns the land and if the owner would let him hunt?(we aren't so very different in that regard,are we)?
BTW,Indians don't hunt birds and are very likely to let you hunt if you can find out who holds the lease.
Don't leave your new truck sitting by the road.You may find it "customized" when you get back.I always take my "beater" truck because it shows modesty and fits-in with the vehicles they drive.