My reading of the manual is that the scores applied to reading, math, and language are within conventional usages. (There is not a better measure of reading on the WIAT-III, due to her grade.) I personally probably would have considered the receptive/expressive vocabulary component scores (probably the higher of the two) from the WIAT-III for the vocabulary measure, since, as you say, the WISC measure has already been included in the ability section, but I can understand the school psych using the WISC-V measure, as it is probably a deeper measure of oral vocabulary than the two achievement measures are (the WIAT-III components are both versions of picture vocabulary, which tends to restrict vocabulary to a more concrete range). For all we know, perhaps that was the highest indicator of vocabulary of the three.
One could make a legitimate argument that the intent of Total Language is either oral language or written language. If the former, then the OL composite is appropriate. If the latter, then the WE composite is appropriate. Again, given her age, choosing OL has a rationale behind it, as the WE composite is purely mechanical at this age (alphabet writing and spelling), and both does not plumb as deep into actual language skills, and also may disadvantage academically-advanced young children with still-developing handwriting skills. (Using OL here probably also explains why measures from this composite were not used for the Vocabulary section; they'd be double-counted.) This probably depends mainly on district policy, though. "Other" is in the same category. If district policy has not defined how Other is used, then the SP is within rights not to apply any score to it, so that candidates are compared to a common yardstick.
Total Achievement combines measures from all aspects of achievement, which has the result either of obscuring focal strengths, or simply duplicating information that has already been entered in the specific academic domains, much like your concern about reusing the WISC-V Vocabulary subtest.
BTW, I assume they administered additional subtests (six beyond the ones you've named) to complete the Aptitude section? At age six, there are age norms for all of the subtests but essay composition and math fluency, and, in any case, they would be using the grade one or two norms for the aptitude measures. If they did, there would be Oral Language, Total Reading, Basic Reading, Written Expression, and Mathematics Composites available for the Aptitude section. You report five composites in addition to Total Achievement.
The school psych's defensiveness, honestly, is irrelevant. (Full disclosure: I'm in the same profession, and have been in similar positions wrt GT decision-making.) Perhaps I would make different professional decisions in his/her case (I can't say, without knowing the specifics), both with regard to scoring and communication, but nothing you've described appears glaringly inappropriate.
