Welcome to Appeals!

117 seems a respectable score (i.e. not too far short of 121).
However, I agree that it's going to be difficult to win a review without the head's support - there would need to be convincing alternative evidence.
What would your evidence be?
Some very high CAT or WISC scores? Predictions of 111+ and GDS for the end of Y6?
https://www.elevenplusexams.co.uk/appea ... cation#b11 As pointed out further above, we don't know what the headteacher recommendation is in your case (is it a '3' or a '4', I wonder?).
We don't know how reliable the school's overall recommendations have been. We caution elsewhere of headteachers who are too generous with their recommendations and therefore lack credibility - but the other extreme could also be possible. In other words, if the head is perceived to be
too strict, this might to some extent compensate for an apparent lack of support, and work to your advantage at a review or appeal.
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DC very committed to school community activities (is this clutching at straws?)
Yes!
I'm afraid it has no relevance at all to the issues that will be considered.
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Extenuating circumstances including a very close family member with advanced cancer going through Chemo in the run up, and DC lives between two different homes (divorced, warring parents). Obviously desperate to got to school with older sibling in what has become a familiar environment.
This sounds sufficient for a score of 117. Might be best presented at an appeal where you would be face to face with the panel, and have the opportunity to respond to any questions about the impact of extenuating circumstances.
The older sibling could also be a useful argument in the event of an appeal where you would probably have to counter oversubscription.