I agree it's best to take the whole lot off. Gravity is your friend here as this gets the air bubbles to float their way out.
I tape the top (calliper) to a shelf, and fit the bleed tube to the calliper and into a clean jar. I then push the fluid into the lever at the bottom (fit a hose union to get a good seal), but keep the syringe upright to ensure any air in the syringe stays out. Above all make sure that the adjusters are wound the right way. The first time it's counter-intuitive the way they go.
The good thing is that you only have to do this job once every 10 years. So overall it's way less time than fiddling with cables, V's and canties.