I've always found screen's ctrl-a is so much easier to reach for than tmux's ctrl-b. I recommend re-mapping ctrl-b to ctrl-a
In any case with your bindings you can still C-z z and it sends C-z to the process.
unbind-key C-b
set -g prefix C-Space
bind-key C-Space send-prefix
I find it a lot easier to type than either Ctrl-A or Ctrl-B.