Unless your "dirty" water is sludgy or contaminated with something you are worried about, don't worry about a little getting in the "clean" water. I usually float the cup for a while to even the temp, and then add a bit of the clean water to the cup 2-3 times over that time, and finally dunk the edge of the cup so the clean water flows in, hold it like that till betta swims out, then remove the cup, which should still contain mostly the old water.
Have a treat ready to tempt him out, I found after the first couple of attempted decuppings that it was either that our pour him out