As it happens, the scoring program(s) that nearly everyone uses for the WISC-V (including in the UK) generates a GAI automatically along with the FSIQ. All of the subtests necessary for the GAI are included in the FSIQ, so no additional testing or even table-lookups are needed (since, unless the psych is in the tiny minority of professionals still hand-scoring, the publisher's software will have done all the work already). Formally, there are not VECI, EFI, EGAI or extended index scores for the UK norms, so those would have to be derived from US norms and interpreted with caution.
But I understand why she does not want to report an FSIQ, as her professional opinion is that it is not a good representation of your child's overall cognitive ability. (In those circumstances, I typically include it in the document somewhere for reference, asterisked, but deemphasize it in my analysis.)
With regard to the GAI, it may be that the spread across just those five subtests is also large enough that she does not judge it to be a good representation of overall cognition. You report that four of the five primary indices were in the Extremely High range. That is not incompatible with a large magnitude of intrasubtest scatter. As a back-of-the-napkin example, consider that a score in the EH range can result from two scaled scores of 16 in the same index. But what actually generates the index score is the sum of the two subtests. So instead of 16, 16, they could have been 13, 19, which is a pretty significant difference. Many evaluators would consider the resulting index score to be a poor representation of the domain, and choose not to report it.
Your child also has a marked relative weakness in processing speed, which may be motor-based, or may be cognitive-based. Or both. Consider that even the GAI includes two timed subtests, which means it can be subject to score-lowering effects in a child with significantly discrepant speed. Consequently, the only index-level score that may not be affected by his known area of weakness is the VCI, and possibly WMI, depending on how weak his fine-motor efficiency is.
Bottom line: there may well be an entirely legitimate professional reason that the evaluator, in her clinical judgement, does not choose to report a composite score (either the FSIQ or the GAI). Have you identiifed a key advocacy use for a formal composite number? If you have, you might try leading with that in your communications with the psych, possibly wrapped in, "I know this may not be the best indicator of his real ability, but it's what the (school, program, etc.) demands in order for him to access this opportunity". It may also be that some resources would respond to presenting the relevant primary index scores for focused advancement.
