How could telling the ship's captain to change the code not
be easy? Maybe Spock was just considering the worst case scenario, but it must be the within the realm of possibility for Khan to change the code. So how hard could it be?
I think we differ in how much intelligence to ascribe to Khan. You think he's super smart, I think he is well above average, but not smart enough to do simple things like read a manual or ask the Captain and First Officer about security measures. Khan's ego is big; is his brain as big as his ego? I don't think so.