I run into that problem alot, if I have to take out a dead one that I think is gonna shatter, I like, jeff said, use a spyder, I have made my own out of misc rope. Sometimes I use the Jameson poles with web slings to choke different parts of the branch, choking only the bigger stuff, clip em together with a biner and then to a large rigging sling, then to the line. Takes a little time and patience! working with web slings and 2 pole together can be frustrating, however, it keeps the big stuff from raining down, I also let the home owner know that I can not stop the itsi bitsi. plywood and tarps also help, I let the ply wood hang over the eve quite a bit, little twigs hitting a aluminum gutter can sound like someone dumped a leader on the house!
I wont let my guys climb deadwood, not worth it, seen enough "solid wood" let go. If HO doesnt like it, hire someone else. My guys go home each night. What we do is dangerous enough,sometimes you just gotta pass.