Scoring 6,5,4 (AKA Ship, Captain, and Crew)
05AB1E, 15 bytes

3ôO&lt;O©3LsK_P®O*


Try it online! or Check out a test suite!

Uses the same trick as Chas Brown and Lynn use: decrementing each integer in each 3x3 sub-matrix rather than subtracting 15 at the end. Expects input in column format.

How it works

3ôO&lt;OD3LsK_PsO* – Full program.
3ô              – Split the input in 3x3 matrices representing dice faces.
  O&lt;O           – Sum each, subtract one and sum each again.
                  This works because after the first O there are 3 elements in each
                  list and there are 5 lists in total, so 3 * 5 = 15 = 4 + 5 + 6.
     ©          – Copy this to the register.
      3LsK      – Push [1, 2, 3] and perform set subtraction. In this scenario, we
                  have already subtracted 3 from each list, so [1, 2, 3] represent
                  the actual dice values [4, 5, 6]. If the resulting list is empty,
                   that means that those values do exist in our roll. Therefore:
          _P    – Produce a list of zeros of that length and take the product
                  (4,5,6 isn't in the dice roll if the list is empty 
                  and this method assures that in this case the product is 1, else 0)
            ®O* – Sum what's in the register and multiply by that.