I would not be going via the Clyde Mtn and Batemans Bay. It's an hour and 100 km longer than going down Brown Mountain. The gradients are around the same on both descents. You can check them out here:

For Brown Mtn see Map of Snowy Mountains Highway, NSW

For Clyde Mtn see Map of Kings Highway

I have no preference for either road, I just take the most convenient one for each trip. I suggest you should take a positive attitude to driving these types of terrain, if you don't, you are going to miss out on a lot of good trips. When you have gained your confidence on this route you will enjoy trips across half a dozen routes across the Great Divide in both NSW and Qld plus a couple in Vic. Just use your gears to make it easy to climb the ascents and to provide engine breaking on the descents. If you slow down and shift into a low enough gear so you don't have to spend too much time on your brakes when descending, you will have good working brakes after you have descended each hill you go down.

                 

I guess it depends upon your level of experience/comfort in towing on hills. I have towed our various caravans up and down the Brown Mountain, Clyde Mountain, Macquarie Pass, and all through the Snowy Mountains (where permitted) on numerous occasions. I used to sometimes drive semi-trailers for a living on these same roads.

Another way is down Mt Darragh. Quite a bit longer to travel to Tathra but the descent is not as long nor steeper.

Drive to Bombala, at the town cross the river and turn left at the roundabout. Follow that road all the way down to the Princes Hwy. While the road is narrower than the Brown as mentioned it is not as long, steep nor winding. At the bottom there is a single lane bridge over a creek that need to be approached with care for oncoming traffic.

(On the way down to Bombala there is a finger sign pointing to Pambula but it is a dirt road and local knowledge was that it was in poor condition. So we opted for sealed roads all the way and ignored it.)

Once on the Princes Hwy turn left and head north to Pambula, Merrimbula or Bega from those towns it is an easy drive to Tathra.

We use the Brown Mountain road approx every 3 weeks in the car and I've driven it dozens of times towing our caravan. It's just a matter of using your gears manually (including with automatic cars), knowing how to brake properly on long descents and having the brake controller set properly. We followed a caravan down there on Monday and the driver clearly did not know this. They were creeping along and riding their brakes the whole way and their brakes were overheating badly.