Turns out there's a larger live limb running parallel to the dead limb and about six feet higher. It's offset just enough to put me in a good position to make a cut halfway out. So, I got two lines over the higher limb, one for the climbing rig, and one for the speed line. I put a good sized stake in the ground to anchor the speed line, and put my climbing rig out on the higher limb and anchored around the tree as usual.
It worked out perfectly. The cut piece landed ten feet from the stake with just a little tug on the limb that I could feel through my harness. Getting the remaining part of the dead limb will take another climb, but I'll be able to use a speedline for it, too. I'll make two cuts to be sure I don't overstress the speedline with the heavier pieces.
Finding another tie-in point and cutting the limb into manageable pieces is the trick.
Thanks for steering me away from a risky setup.
About my climbing rig: It's a 2:1 system with a ball-bearing pulley at the top. A rope runs from the top of the pulley over the limb and down to the basal anchor. I use a Zigzag and a double handed ascender with a footloop to go up, and the ZZ to come down. If I was a lot younger, I'd be using something like you-all are, but this works really well for me now.