Of the ten Remote bush camps in Lamington National Park Echo Point, Lost World Creek, Lost World Saddle, Mount Bithongabel, Point Lookout, Rat-A-Tat, Running Creek and Stinson are all in the Green Mountains section and south along the range. Darlington is situated on the Darlington Range which is between Binna Burra and o’reilly’s which is accessed from gwongoorool pool along the Illinbah circuit. The Illinbah camp is half way around the Illinbah circuit. Realistically none of the bushcamps in Lamington are going to be helpful to you doing the ship stern circuit.
The Woonoongoora walkers’ camp along the great walk departs on the same Eastern side of Binna Burra though is 19.5km each way and would just be adding an out and back hike and wouldn’t be particuarly helpful for trying to turn the ship stern circuit into an over night hike.
You could try to use the border track to Bithongabel campsite stay the night then on day two return along the track and branch of and do the ship stern circuit returning to binna burra which would mean you did a very small bit of the circuit on the first day a long detour to camp and then back and then complete the rest of the circuit the rest being almost the entirety of the ship stern circuit bar the small border track section so would make a lengthy second day.
Which if your point of trying to make it an overnighter was to have shorter days this would definetly be counter productive as would trying to link in any of the other bushcamps as they are all located significantly out of the way. There is probably no way for you to camp around the ship stern circuit as a way to break up the journey though you can camp at binna burra itself though that’s essentially just doing the ship stern circuit as a day walk and camping at binna burra as it’s the start point / end point for the walk. If you just wanted a overnight circuit walk from Binna burra the Illinbah Circuit is more designed for that. Where as the ship stern would logistically be better as a day walk and you could spend night before / after at binna burra.