A Terminator plugin for 1-click dumping of the console contents to a text file (~/.terminator/$datetime.log).
One needs to copy the plugin to /usr/share/terminator/terminatorlib/plugins/ or ~/.config/terminator/plugins/ and enable it under Preferences to make the context-menu (named DumpToFile) appear. Dump files are stored under ~/.terminator.
For Gtk3 versions of Terminator (ver 1.90+) use the terminator-1.90 branch here.