My apologies for the delay, Joe, I was on vacation. Now that I'm home, I gave this a try but at least on my machine writing to /dev/shm/ works as I remembered, even with --read-only:

    $ docker run --read-only -ti ubuntu bash
    root@3dfdab770505:/# echo "bar" > /dev/shm/foo
    root@3dfdab770505:/# cat /dev/shm/foo
    bar

So, again, couldn't you just write your binary to /dev/shm and execute it?