I know what these kids go through. I was that kid myself. I know I was very aware that I wasn't "normal". You're correct that in the age range 4-7 a child wouldn't know to give the "expected" answer. It won't be much longer though until that child can use his/her intelligence to work out the expected response. At my school, it wasn't socially acceptable to show intelligence so maybe I learnt to hide it earlier than most.
I don't know my IQ. I've had to take tests a few times as an adult for work and every time I land "most likely somewhere in the top 0.1%" according to the psychologists. I don't really feel it matters if my tests show 145 or 160. A large school with 800 kids is only going to have a single child in that range.
You can either move that child up some years, or you can give them a customised workload. A child that is 180 IQ is very different from a child that's 145 but the approach of attempting to find them an appropiate workload would be a good start.
Using it wasn't intended to suggest a classroom setting for the story in question, just a standard answer that someone more knowledgeable about history might not spit out as The Answer without being inculcated to do so.
This was an actual anecdote from a discussion with a parent of the child in question. The child in question was probably between four and seven years old. They were being assessed by a professional one on one, not in a classroom setting. They were neither showing off nor in possession of the social skills you assume they surely had to have.
I will add that imposter syndrome is common among adults because it's quite hard to figure out where we stand in relation to other people, so smart adults frequently feel they must be "average" because this is "normal" for them. Children are at substantial disadvantage compared to adults.
Do you need to know if it's 145 verse 155?
Possibly. The standard deviation for IQ tests is just 15 points.
More importantly, a test that only goes to 145 cannot distinguish 145 from 180. Both will likely score 145 or close to it.