Jim, I actually didn't lose any beaugainvillas. Just many years worth of growth.
We cut all the dead branches down to the stems and they sprouted nicely from scratch. Just no flowers. I had several plants trained on a wire trellis and these exposed branches all died. New stuff is coming though. Should be OK next year for a nice bloom. I also have one trained across three palms and after we trimmed and wounded ourselves it's looking good again. This picture is past the major flowering I have better shots somewhere but you get the idea.