I would query the school and school district about what services and supports would be available, and find out what testing they would do, and under what circumstances. Where I am, testing is around $2k - not cheap.
The truth is that many schools do relatively little to nothing for gifted students. There may be some lip service about in-class differentiation, or occasional pullouts. I base this on our experience, those of other families we know locally, reading here, in books and elsewhere online. Some areas do offer gifted schools, gifted classes, structured pullouts etc. I would talk to the school as a starting point.
DS was tested at 5 and again at almost 8. Both told different stories - DS's VS skills were very strong at 5, while his verbal skills were very strong at 8. The testing at 5 was not "clean" as he is a very divergent thinker and is 2E (twice exceptional - gifted with learning differences). In retrospect, I wouldn't have tested at 5 because we came out of it knowing we'd need to test again in the future, and other than confirming he's gifted, it didn't particularly benefit him.
There is evidence that IQ is pretty variable - swinging up to 30 points either way through the childhood years. According to Nurtureshock:
Warning: Spoiler! (Click to show)
if you picked 100 kindergartners as "gifted," by third grade only 27 of them would still deserve that categorisation. You would have loocked out 73 other deserving students.
Some kids are late bloomers, some are highly verbal which is predominantly what's noticed at early ages... IQ testing is not considered "reliable" until 3rd grade.
OP, if you think it would gain your child access to useful services but the school won't do it, and you think your son would enjoy it, I would go for it. A child with your son's abilities is pretty likely gifted and some schools are going to need paper to document it. My son thought the testing sessions were fun. Kindie is fun, grade 1 can be a different story as they increase focus on "learning to read" and there's less time for play.
These topics have been covered many times in this forum so you might enjoy reading back for some interesting perspectives and experiences.