Long Way To Go
Listen here "warrior", I'm currently serving in a conventional infantry unit myself (2nd ID, 3rd Stryker BDE), and also making arrangements to get back to my beloved airborne community. That being said...if it takes a lot of painful coordination and approval signatures for a decorated E-6 paratrooper with a 296-310 APFT average to do that kind of reassignment on Fort Lewis, then you as a Private are probably in over your head. Especially when it comes to requesting a transfer to a high speed unit like that. If I were your First Sergeant and you asked for such a thing, I would make you do the caterpillar crawl out of my office with your legs tied together and a sign taped to your back that reads "Larva Needs Silver Wings"
Even on the off chance that you made it in the regiment, they would squeeze your deluded self-image into virgin juice and tell the Extra Duty maggots to use it for watering the parade field.
In other words, you probaby wouldn't even last a day in my current squad with all the other legs that are fearfully learning the meaning of airborne disciple. Take the advice that those before me have already recommended and go to Jump School, work on PT, and get a few deployments under your belt. Face the reality, leaders don't want to put up with fresh fish, it seems your only quality is to be used as a detail pawn right now.
P.S. Do your own research before asking people questions, that's a life Army Lesson for you.
Oh and good luck in your career moves.