MBB--Interesting problem and one I have pondered a bit. You don't mention which streams you have in mind but the most obvious ones are the Park portions of Bunches Creek and Raven Fork. Since I never fish the Reservation waters I'm not sure if the "open to the public with permits" fishing on either goes all the way to the Park line, but a look at the free map Tribal fisheries provides would tell you. There are some streams on the Reservation set aside solely for tribal members, and I tend to think that Raven Fork at the point where it adjoins the Park is one of them. I know that is the case with Straight Fork from its juncture with Raven Fork to the Park line.
You might want to try to talk to Robert Blankenship or the fellow who heads the Tribal Fisheries program (can't recall his name off the top of my head) and see what they say. Frankly, I'd also worry a bit about leaving a vehicle in the two places I mention above.
Finally, if you do figure out a way to get into Raven Fork at the Park boundary, be aware of the fact that you are entering really remote and rugged country. You will be in "The Gorges" in fairly short order, and there is no place anywhere in the Park which is tougher going. I definitely wouldn't advise doing it by yourself, and unless you are reasonably fit and a lot more "catty" than this 68-year-old, don't do it at all. There are places--lots of them--where the only real way to progress upstream is to swim. The banks on either side are that steep and some of the pools are over your head.
Hope this helps a bit, and if you pursue it, I'd love to know what you learn. There is, I might add, another way into Bunches Creek (although the way back out isn't exactly easy).