I grew up in Atlanta area and North GA and started boating there and in the TN, NC, SC boating rivers and creeks. Moved west a number of years ago but still go back to the area you mentioned to have fun.
Not a lot of oar rigs used by private boaters in that area of the world. Mostly kayaks, canoes and in rafts all the commercials use paddle rafts. There are some private boaters that do R2 in various inflatables.
From what you posted, I agree the Ocoee might work for you. The shuttle of Section 4 of the Chatooga is pretty time consuming. Or used to be. Check shuttles with NOC.
My experience driving from Atlanta to the Ocoee is it takes more than a couple hours, your experience may vary. The traffic load getting from the Ocoee down and thru to the I75 interstate South to Atlanta can be one long traffic jam on weekends.
The normal run of the Ocoee runs right beside the highway. You can scout most of the river from the highway while driving up to the normal put in.
The Ocoee is one of the most enjoyed river runs in the US. Right up there people wise counts with the ARK in Colorado. So expect crowds especially on the week ends. Rapids may not be hardest Idaho grade but can be challenging to most of us. SE boaters and Idaho boaters seem to rate runs and rapids differently at least in my experience.
A suggestion is run the Ocoee with a outfitter in a paddle raft for your first run. This will show you the lines. Then you are prepared with the lines that work and provide a lot of excitement. The normal run of the Ocoee is only a few hours and I have made multiple runs in one day.
On weekends as a private boater usually in a kayak, my experience is at the play spots expect a pretty much continual line of commercial paddle rafts most of which will run over any thing in their path. So privates will dart in and out to surf etc. Just be prepared.
NOC is one of the nationally known outfitters. They run almost all of the classic SE runs commercially usually in paddle rafts but some in kayaks or canoes. They also may do shuttles if asked. I use and have used NOC for all my boating career and they have always given me good advice and service on shuttles, places to eat, rooms and clinics. Give them a call is my recommendation.
Endless Rivers outfitters is head quartered just up the road from NOC on the Nanty. They provide quality of service at the highest level as well. I recommend you call them as well as NOC and describe what you are looking for. They run all the SE classics as well and do boutique style outfitting as well.
NOC has trained quite a few Olympic grade boaters and provided clinics for uncounted many more private boaters. Most SE boaters will start on the Nantahala then move up to the Ocoee for real training on real rapids. Some will rate the Ocoee normal run as class 4 and some a hard 3. Regardless of rating I find the Ocoee to be a super fun run and you can make the lines harder or easier, but you need to know the lines. Water is usually a bit cold even in summer time. Traffic between the normal put in and take out will be constant on week end with vehicles and people hitch hiking between the put in take out. Be aware that the TN and NC law enforcement knows the traffic load in and out of the Ocoee area. They enforce their laws so be advised to follow their rules.
Others have listed awesome runs as well in NC and other parts of the SE. My advice is the Ocoee is probably the best for the time you have to spend. For sure it will be one of the easiest to get to and from Atlanta in the time frame you mentioned.
All in all the Ocoee is a fun run. Enjoy but be safe. Check the run out and make your own decisions on how to run it. The Ocoee is a really fun private float in week. Do the week end commercial float and you may well change your travel plans and go back in week when crowds are much less. My advice is just that, take a look at the runs and make your own decision on what to do.