I'm a noob here. Both to the forum and to cycling. But... i actually live in Brunswick, the mainland side of SSI. I really like biking on Jekyll too. As mentioned above, SSI is packed, especially this time of year, and Jekyll is deserted in comparison. There are some good trails on Jekyll and road riding is good as the traffic is almost non-existent. I think the mentioed trip from SSI lighthouse to Jekyll and back might be enjoyable if you are looking for a ride of that distance.
The bridge to jekyll does have a sizeable shoulder and I see folks all the time biking it.
good luck