I have never been able to root a 3" cutting, and I have rooted hundreds of cuttings. Others say they can do it, but I just can't.
Keep in mind that plumeria without leaves, and certainly without roots, can't utilize water or fertilizer. Rot may be occurring because you are keeping that potting soil moist and rot occurs. I don't root my cuttings in soil, but bag them. For years I rooted mine in 100% coarse perlite and that perlite was bone dry. Bagging is a different philosophy, but I seem to get faster rooting with the bagging method. During rooting, the plant needs to stay warm (70 - 95 F is best). It really doesn't matter whether there is bright or shaded light, but no direct sunlight. Are you using a rooting hormone, which has a fungicide to counter rot?