"Doesn't IQ vary over time though"
Children obviously become cleverer with age i.e. 10yo DD can do things 1yo DD couldn't. IIRC it's around age 25 when we peak and it all starts going downhill again. But of course IQ doesn't care about that because it is a relative ranking amongst peers of the same age. IQ becomes fairly stable with increased age. Here's a line in a paper where one co-author, Deary, is quite a big name in intelligence research:
"The correlation between the Binet IQ at age 11 and verbal ability at age 77 was .72. Adding on a group-administered test at age 11 and educational information raised the multiple R to > .8."
Can't find it quickly, but I've seen other research saying adult IQs e.g. aged 20 vs. 50, have a very strong correlation.
"Its not like it could enable them to pass their 11+ or anything"
All those tests have VR/NVR don't they? If this was a top 25% grammar area I'd have thrown some practice at DD anyway. Just in case. You never know. They might panic or have the sniffles on the day and it could save them.
Perhaps I'm more inclined to think my way because we live on the edge of a very large catchment area for a shiny school that does (partial) super-selection and also has music places. It's 3+hours travel per day, but if it were round the corner I think I'd have been compelled to try and level the playing field, drill her quite a bit on the academic test-stuff, turn the hour per-week piano practice into a similar amount every day etc. Threaten to take all her stuffed animals and burn them. All despite not knowing if DD was a genuinely credible contender. Possibly not the Chua part, but several of her whizzier new friends at the comp did go through that.