Couple of considerations for open waters (by Park standards): (1) West Prong Little Pigeon past Sugarlands on US 441 at Quiet Walkways or (2) Little River just above Elkmont CG.
Inside the Park you need only a TN (or NC) license. TN Trout stamp not required.
Since your time is limited, LRO may be out of reach, in which case I suggest you drop in at The Smoky Mountain Angler
located on Brookside Village Way riight off US 321 just a short distance from the 321/441 split in G'burg. There you can get tips on flies, and any needed merchandise.
Enjoy your time.