I'm only working at postgrad level so yes, I guess you're right I do have to produce meaningless tests to satisfy him/her. But if that's what gets me a grade and to pass an assignment, it's just something I have to get on with unfortunately. I don't know if they have a clue about statistics or not, but I know they're giving me a grade so.

No! No, no no!

Don't do this! This is the attitude of someone who lives in a dictatorship.

Science: "This is true because the evidence says so".

Non-scientific: "This is true because may boss says so."

Stand up! Straighten your body and show you have a spine! Question authority! If you just do what your boss tells you, then you have nothing to be proud of when you have your exam. Instead, show the evidence!

I had to calculate a sum of scores for each measure,

So you have already calculated a sum. Then, remember, the central limit theorem tells you that a sum of random variables will be approximately normally distributed. So 1) you have theoretic evidence to believe that it will be roughly normal.

Second 2) you have shown the histograms above. The histograms shows that the data are bell-shaped, roughly normal.

Third 3) the tests that are based on the normal distribution are robust for minor non-normality. Even for the very skewed exponential distribution the test are fairly robust if the sample size is larger than n=30. (And from the histograms the data seems to be somewhat larger than n=30)

So just tell your software to display the p-values associated with the Shapiro-Wilks test statistics, and report them.

George Box, a great statistician, has described this procedure as going out with a small rowing boat to check the waves, to see if a big Atlantic steamer dares to leave the port. The rowing boat will discover the smallest waves. The steamer will not notice them. the Shapiro-Wilks-test will discover the smallest deviation from normality. A t-test will not notice (and not be affected).

Sometimes you need to throw something to the wolves. Here are the p-values for the Shapiro-Wilks. But that is irrelevant. For your self you can think to put it where Ernst Kuzorra, 1941, suggested someone to do something. (»Jetzt kann mich der Führer mal am Arsch lecken.« (Ernst Kuzorra, 1941))

You can also look at normality via q-q plots.

Yes, that is a good graphical method. But you have already shown the histograms. And they are convincing enough.

Now, stand up! And be proud!