Hey there am4mu, just registered on here recently and just saw your post. I grew up in the Ozarks and moved back last year, but have spent the last ten years living literally about 4 minutes from the park boundary and 20 minutes from Gatlinburg. I spent that time fishing all over the park, outside of it, and Western North Carolina. I might be able to offer up some helpful info. on getting the most out of your time there.
First off, the great thing about the Smokies area is that you have so many choices in the way you prefer to fish on a given day based on particular preference or mood. (tailwater, big stockers, wild rainbows, up high brookies, backcountry trout with total seclusion, lake trout and Smallmouth from kayak, a quick trip near the road, wading for smallmouth, trout and redeye outside the park boundary, a daypack trip to a seldom fished stream) etc. The neat thing is because of proximity you can choose at the last minute and still get it done. So with all that being said, my recommendation would be to not limit yourself to either or. The only thing your missing by not fishing Gatlinburg though is the opportunity to catch much bigger than average fish for the area (stockers), and you can do that in a much prettier setting when you get back here. I say set aside about an hour and a half to take in the spectacle that is Gatlinburg and then go enjoy what the Smokies are all about, which is unspoiled natural beauty and the abundant history surrounding the appalachian settlers that were able to carve out an existance there.
Someone mentioned Abram's Creek earlier. You guys will definitely want to see Cade's Cove while you are there, which is how you access upper Abram's but I wouldn't send someone there if they were budgeting fishing time as efficiently as possible. My favorite area to fish on that creek is an area called the "horseshoe", but you need a full day to fish it and it is rugged as all get out to get in and out of. (hence the good quality fishing). Search and rescue goes in there every couple years for someone that doesn't give themselves enough time in the day to fish it through and then trys to bushwack out in the dark. The rest of the creek is right next to the always busy Abram Falls trail and the wading is as slick as it gets in the entire park. You can do better with the time you have, but if you've got your gear give it a go anyway since you'll be right there.
The fly shop mentioned is in Townsend and is called Little River Outfitters. They built a new building a few years ago and it is owned by Byron and Paula Begley and Daniel Drake. They are great people and the shop is as nice as any you will ever visit. If you go to their website and go to the message board it would be very helpful in getting familiar with the surrounding area and local streams.
I will end this lengthy post with just a couple more things. First, I will say that if I were heading back tomorrow and had only one day to fish, I would be thinking of one thing and one thing only....grabbing my 3 weight and heading up high to match wits with native brookies on a dry fly. The fish are small (4-6") on average, but the locale, fishing technique required and overall experience are second to none and I guarantee will be memorable. I miss that more than I thought I would. Secondly, as long as you keep your gear in your vehicle as you and your wife sightsee, you will find more trout-holding water than you could fish in years. It's the trout mecca of the East IMO. PM me for more specific info. if you would like.