You can always build a simple bubble cloner out of a small rubbermaid box, a couple air stones & an aquarium air pump. The idea is that you just punch a few holes in the lid of the container for the cuttings, put the airstones in the bottom of the container and it oxygenates the water preventing the need for changing it. Also, as the water evaporates the level drops below the tip of the cutting, leaving the part thats forming roots in a humid but not submerged environment encouraging faster rooting. Works great and costs under 20 bucks to build. Say good bye to rooting hormone too, all you need is plain water.
I'll post some pics of the one i built. So far it has worked for tomatoes, stevia, & salvia and a couple other plants. I did a test run with stevia, 1 rooting in water, 1 rooting in perlite, and 1 rooting in the cloner. Cloner had much bigger roots and rooted about 3-4 days earlier than the other methods, even though I used rooting hormone on the perlite one.
The reason I mention all this is because normally to clone something you must cut off most of the leaves to prevent moisture loss due to transpiration, but with this method it seems to work on much larger cuttings without wilting or the need for a humidity dome. So maybe it would work for your big branch.
Also, one trick I learned from a florist is to make the cut on the stem while holding it underwater if possible. This prevents an air bubble from getting inside the stem which would impede the further uptake of water, thus preventing wilting. Try that if you ever have problems with flowers or cuttings wilting even if you put them in water.