피곤해서 그래 is the correct, kind version. 피곤하거든 sounds way too serious and stressed, thus unkind. 피곤하거든 can also mean "if" 피곤하거든 방에 가서 자. If you are tired, go to your room and sleep 피곤하거든? might sound very rude, something like "Don't you see that I'm tired?"

Here's some probable conversation scenarios. A: 괜찮아? B: 응, 괜찮아. / 응, 근데 좀 피곤해 / 아니, 피곤해서 힘들어. A: 힘들어 보인다 [안 좋아 보인다]. 왜 그래? B: 응, 좀 피곤하거든 / 응, 피곤해서 그래 / 어제 잠을 못잤거든 / 어제 잠을 못자서 그래. ~해서 그래 is "It is (so) because ...". ~거든 is like part giving a reason and part offering new information. ~거든 has other uses too, but using it to give a reason or background like the above is fine.